<!DOCTYPE html><html><head>
      <title>07</title>
      <meta charset="utf-8">
      <meta name="viewport" content="width=device-width, initial-scale=1.0">
      
      <style>
      /*!
 * reveal.js
 * http://revealjs.com
 * MIT licensed
 *
 * Copyright (C) 2018 Hakim El Hattab, http://hakim.se
 */
/*********************************************
 * RESET STYLES
 *********************************************/
html, body, .reveal div, .reveal span, .reveal applet, .reveal object, .reveal iframe,
.reveal h1, .reveal h2, .reveal h3, .reveal h4, .reveal h5, .reveal h6, .reveal p, .reveal blockquote, .reveal pre,
.reveal a, .reveal abbr, .reveal acronym, .reveal address, .reveal big, .reveal cite, .reveal code,
.reveal del, .reveal dfn, .reveal em, .reveal img, .reveal ins, .reveal kbd, .reveal q, .reveal s, .reveal samp,
.reveal small, .reveal strike, .reveal strong, .reveal sub, .reveal sup, .reveal tt, .reveal var,
.reveal b, .reveal u, .reveal center,
.reveal dl, .reveal dt, .reveal dd, .reveal ol, .reveal ul, .reveal li,
.reveal fieldset, .reveal form, .reveal label, .reveal legend,
.reveal table, .reveal caption, .reveal tbody, .reveal tfoot, .reveal thead, .reveal tr, .reveal th, .reveal td,
.reveal article, .reveal aside, .reveal canvas, .reveal details, .reveal embed,
.reveal figure, .reveal figcaption, .reveal footer, .reveal header, .reveal hgroup,
.reveal menu, .reveal nav, .reveal output, .reveal ruby, .reveal section, .reveal summary,
.reveal time, .reveal mark, .reveal audio, .reveal video {
  margin: 0;
  padding: 0;
  border: 0;
  font-size: 100%;
  /* font: inherit; */
  vertical-align: baseline; }

.reveal article, .reveal aside, .reveal details, .reveal figcaption, .reveal figure,
.reveal footer, .reveal header, .reveal hgroup, .reveal menu, .reveal nav, .reveal section {
  display: block; }

/*********************************************
 * GLOBAL STYLES
 *********************************************/
html,
body {
  width: 100%;
  height: 100%;
  overflow: hidden; }

body {
  position: relative;
  line-height: 1;
  background-color: #fff;
  color: #000; }

/*********************************************
 * VIEW FRAGMENTS
 *********************************************/
.reveal .slides section .fragment {
  opacity: 0;
  visibility: hidden;
  transition: all .2s ease; }
  .reveal .slides section .fragment.visible {
    opacity: 1;
    visibility: inherit; }

.reveal .slides section .fragment.grow {
  opacity: 1;
  visibility: inherit; }
  .reveal .slides section .fragment.grow.visible {
    -webkit-transform: scale(1.3);
            transform: scale(1.3); }

.reveal .slides section .fragment.shrink {
  opacity: 1;
  visibility: inherit; }
  .reveal .slides section .fragment.shrink.visible {
    -webkit-transform: scale(0.7);
            transform: scale(0.7); }

.reveal .slides section .fragment.zoom-in {
  -webkit-transform: scale(0.1);
          transform: scale(0.1); }
  .reveal .slides section .fragment.zoom-in.visible {
    -webkit-transform: none;
            transform: none; }

.reveal .slides section .fragment.fade-out {
  opacity: 1;
  visibility: inherit; }
  .reveal .slides section .fragment.fade-out.visible {
    opacity: 0;
    visibility: hidden; }

.reveal .slides section .fragment.semi-fade-out {
  opacity: 1;
  visibility: inherit; }
  .reveal .slides section .fragment.semi-fade-out.visible {
    opacity: 0.5;
    visibility: inherit; }

.reveal .slides section .fragment.strike {
  opacity: 1;
  visibility: inherit; }
  .reveal .slides section .fragment.strike.visible {
    text-decoration: line-through; }

.reveal .slides section .fragment.fade-up {
  -webkit-transform: translate(0, 20%);
          transform: translate(0, 20%); }
  .reveal .slides section .fragment.fade-up.visible {
    -webkit-transform: translate(0, 0);
            transform: translate(0, 0); }

.reveal .slides section .fragment.fade-down {
  -webkit-transform: translate(0, -20%);
          transform: translate(0, -20%); }
  .reveal .slides section .fragment.fade-down.visible {
    -webkit-transform: translate(0, 0);
            transform: translate(0, 0); }

.reveal .slides section .fragment.fade-right {
  -webkit-transform: translate(-20%, 0);
          transform: translate(-20%, 0); }
  .reveal .slides section .fragment.fade-right.visible {
    -webkit-transform: translate(0, 0);
            transform: translate(0, 0); }

.reveal .slides section .fragment.fade-left {
  -webkit-transform: translate(20%, 0);
          transform: translate(20%, 0); }
  .reveal .slides section .fragment.fade-left.visible {
    -webkit-transform: translate(0, 0);
            transform: translate(0, 0); }

.reveal .slides section .fragment.fade-in-then-out,
.reveal .slides section .fragment.current-visible {
  opacity: 0;
  visibility: hidden; }
  .reveal .slides section .fragment.fade-in-then-out.current-fragment,
  .reveal .slides section .fragment.current-visible.current-fragment {
    opacity: 1;
    visibility: inherit; }

.reveal .slides section .fragment.fade-in-then-semi-out {
  opacity: 0;
  visibility: hidden; }
  .reveal .slides section .fragment.fade-in-then-semi-out.visible {
    opacity: 0.5;
    visibility: inherit; }
  .reveal .slides section .fragment.fade-in-then-semi-out.current-fragment {
    opacity: 1;
    visibility: inherit; }

.reveal .slides section .fragment.highlight-red,
.reveal .slides section .fragment.highlight-current-red,
.reveal .slides section .fragment.highlight-green,
.reveal .slides section .fragment.highlight-current-green,
.reveal .slides section .fragment.highlight-blue,
.reveal .slides section .fragment.highlight-current-blue {
  opacity: 1;
  visibility: inherit; }

.reveal .slides section .fragment.highlight-red.visible {
  color: #ff2c2d; }

.reveal .slides section .fragment.highlight-green.visible {
  color: #17ff2e; }

.reveal .slides section .fragment.highlight-blue.visible {
  color: #1b91ff; }

.reveal .slides section .fragment.highlight-current-red.current-fragment {
  color: #ff2c2d; }

.reveal .slides section .fragment.highlight-current-green.current-fragment {
  color: #17ff2e; }

.reveal .slides section .fragment.highlight-current-blue.current-fragment {
  color: #1b91ff; }

/*********************************************
 * DEFAULT ELEMENT STYLES
 *********************************************/
/* Fixes issue in Chrome where italic fonts did not appear when printing to PDF */
.reveal:after {
  content: '';
  font-style: italic; }

.reveal iframe {
  z-index: 1; }

/** Prevents layering issues in certain browser/transition combinations */
.reveal a {
  position: relative; }

.reveal .stretch {
  max-width: none;
  max-height: none; }

.reveal pre.stretch code {
  height: 100%;
  max-height: 100%;
  box-sizing: border-box; }

/*********************************************
 * CONTROLS
 *********************************************/
@-webkit-keyframes bounce-right {
  0%, 10%, 25%, 40%, 50% {
    -webkit-transform: translateX(0);
            transform: translateX(0); }
  20% {
    -webkit-transform: translateX(10px);
            transform: translateX(10px); }
  30% {
    -webkit-transform: translateX(-5px);
            transform: translateX(-5px); } }
@keyframes bounce-right {
  0%, 10%, 25%, 40%, 50% {
    -webkit-transform: translateX(0);
            transform: translateX(0); }
  20% {
    -webkit-transform: translateX(10px);
            transform: translateX(10px); }
  30% {
    -webkit-transform: translateX(-5px);
            transform: translateX(-5px); } }

@-webkit-keyframes bounce-down {
  0%, 10%, 25%, 40%, 50% {
    -webkit-transform: translateY(0);
            transform: translateY(0); }
  20% {
    -webkit-transform: translateY(10px);
            transform: translateY(10px); }
  30% {
    -webkit-transform: translateY(-5px);
            transform: translateY(-5px); } }

@keyframes bounce-down {
  0%, 10%, 25%, 40%, 50% {
    -webkit-transform: translateY(0);
            transform: translateY(0); }
  20% {
    -webkit-transform: translateY(10px);
            transform: translateY(10px); }
  30% {
    -webkit-transform: translateY(-5px);
            transform: translateY(-5px); } }

.reveal .controls {
  display: none;
  position: absolute;
  top: auto;
  bottom: 12px;
  right: 12px;
  left: auto;
  z-index: 1;
  color: #000;
  pointer-events: none;
  font-size: 10px; }
  .reveal .controls button {
    position: absolute;
    padding: 0;
    background-color: transparent;
    border: 0;
    outline: 0;
    cursor: pointer;
    color: currentColor;
    -webkit-transform: scale(0.9999);
            transform: scale(0.9999);
    transition: color 0.2s ease, opacity 0.2s ease, -webkit-transform 0.2s ease;
    transition: color 0.2s ease, opacity 0.2s ease, transform 0.2s ease;
    z-index: 2;
    pointer-events: auto;
    font-size: inherit;
    visibility: hidden;
    opacity: 0;
    -webkit-appearance: none;
    -webkit-tap-highlight-color: transparent; }
  .reveal .controls .controls-arrow:before,
  .reveal .controls .controls-arrow:after {
    content: '';
    position: absolute;
    top: 0;
    left: 0;
    width: 2.6em;
    height: 0.5em;
    border-radius: 0.25em;
    background-color: currentColor;
    transition: all 0.15s ease, background-color 0.8s ease;
    -webkit-transform-origin: 0.2em 50%;
            transform-origin: 0.2em 50%;
    will-change: transform; }
  .reveal .controls .controls-arrow {
    position: relative;
    width: 3.6em;
    height: 3.6em; }
    .reveal .controls .controls-arrow:before {
      -webkit-transform: translateX(0.5em) translateY(1.55em) rotate(45deg);
              transform: translateX(0.5em) translateY(1.55em) rotate(45deg); }
    .reveal .controls .controls-arrow:after {
      -webkit-transform: translateX(0.5em) translateY(1.55em) rotate(-45deg);
              transform: translateX(0.5em) translateY(1.55em) rotate(-45deg); }
    .reveal .controls .controls-arrow:hover:before {
      -webkit-transform: translateX(0.5em) translateY(1.55em) rotate(40deg);
              transform: translateX(0.5em) translateY(1.55em) rotate(40deg); }
    .reveal .controls .controls-arrow:hover:after {
      -webkit-transform: translateX(0.5em) translateY(1.55em) rotate(-40deg);
              transform: translateX(0.5em) translateY(1.55em) rotate(-40deg); }
    .reveal .controls .controls-arrow:active:before {
      -webkit-transform: translateX(0.5em) translateY(1.55em) rotate(36deg);
              transform: translateX(0.5em) translateY(1.55em) rotate(36deg); }
    .reveal .controls .controls-arrow:active:after {
      -webkit-transform: translateX(0.5em) translateY(1.55em) rotate(-36deg);
              transform: translateX(0.5em) translateY(1.55em) rotate(-36deg); }
  .reveal .controls .navigate-left {
    right: 6.4em;
    bottom: 3.2em;
    -webkit-transform: translateX(-10px);
            transform: translateX(-10px); }
  .reveal .controls .navigate-right {
    right: 0;
    bottom: 3.2em;
    -webkit-transform: translateX(10px);
            transform: translateX(10px); }
    .reveal .controls .navigate-right .controls-arrow {
      -webkit-transform: rotate(180deg);
              transform: rotate(180deg); }
    .reveal .controls .navigate-right.highlight {
      -webkit-animation: bounce-right 2s 50 both ease-out;
              animation: bounce-right 2s 50 both ease-out; }
  .reveal .controls .navigate-up {
    right: 3.2em;
    bottom: 6.4em;
    -webkit-transform: translateY(-10px);
            transform: translateY(-10px); }
    .reveal .controls .navigate-up .controls-arrow {
      -webkit-transform: rotate(90deg);
              transform: rotate(90deg); }
  .reveal .controls .navigate-down {
    right: 3.2em;
    bottom: 0;
    -webkit-transform: translateY(10px);
            transform: translateY(10px); }
    .reveal .controls .navigate-down .controls-arrow {
      -webkit-transform: rotate(-90deg);
              transform: rotate(-90deg); }
    .reveal .controls .navigate-down.highlight {
      -webkit-animation: bounce-down 2s 50 both ease-out;
              animation: bounce-down 2s 50 both ease-out; }
  .reveal .controls[data-controls-back-arrows="faded"] .navigate-left.enabled,
  .reveal .controls[data-controls-back-arrows="faded"] .navigate-up.enabled {
    opacity: 0.3; }
    .reveal .controls[data-controls-back-arrows="faded"] .navigate-left.enabled:hover,
    .reveal .controls[data-controls-back-arrows="faded"] .navigate-up.enabled:hover {
      opacity: 1; }
  .reveal .controls[data-controls-back-arrows="hidden"] .navigate-left.enabled,
  .reveal .controls[data-controls-back-arrows="hidden"] .navigate-up.enabled {
    opacity: 0;
    visibility: hidden; }
  .reveal .controls .enabled {
    visibility: visible;
    opacity: 0.9;
    cursor: pointer;
    -webkit-transform: none;
            transform: none; }
  .reveal .controls .enabled.fragmented {
    opacity: 0.5; }
  .reveal .controls .enabled:hover,
  .reveal .controls .enabled.fragmented:hover {
    opacity: 1; }

.reveal:not(.has-vertical-slides) .controls .navigate-left {
  bottom: 1.4em;
  right: 5.5em; }

.reveal:not(.has-vertical-slides) .controls .navigate-right {
  bottom: 1.4em;
  right: 0.5em; }

.reveal:not(.has-horizontal-slides) .controls .navigate-up {
  right: 1.4em;
  bottom: 5em; }

.reveal:not(.has-horizontal-slides) .controls .navigate-down {
  right: 1.4em;
  bottom: 0.5em; }

.reveal.has-dark-background .controls {
  color: #fff; }

.reveal.has-light-background .controls {
  color: #000; }

.reveal.no-hover .controls .controls-arrow:hover:before,
.reveal.no-hover .controls .controls-arrow:active:before {
  -webkit-transform: translateX(0.5em) translateY(1.55em) rotate(45deg);
          transform: translateX(0.5em) translateY(1.55em) rotate(45deg); }

.reveal.no-hover .controls .controls-arrow:hover:after,
.reveal.no-hover .controls .controls-arrow:active:after {
  -webkit-transform: translateX(0.5em) translateY(1.55em) rotate(-45deg);
          transform: translateX(0.5em) translateY(1.55em) rotate(-45deg); }

@media screen and (min-width: 500px) {
  .reveal .controls[data-controls-layout="edges"] {
    top: 0;
    right: 0;
    bottom: 0;
    left: 0; }
  .reveal .controls[data-controls-layout="edges"] .navigate-left,
  .reveal .controls[data-controls-layout="edges"] .navigate-right,
  .reveal .controls[data-controls-layout="edges"] .navigate-up,
  .reveal .controls[data-controls-layout="edges"] .navigate-down {
    bottom: auto;
    right: auto; }
  .reveal .controls[data-controls-layout="edges"] .navigate-left {
    top: 50%;
    left: 8px;
    margin-top: -1.8em; }
  .reveal .controls[data-controls-layout="edges"] .navigate-right {
    top: 50%;
    right: 8px;
    margin-top: -1.8em; }
  .reveal .controls[data-controls-layout="edges"] .navigate-up {
    top: 8px;
    left: 50%;
    margin-left: -1.8em; }
  .reveal .controls[data-controls-layout="edges"] .navigate-down {
    bottom: 8px;
    left: 50%;
    margin-left: -1.8em; } }

/*********************************************
 * PROGRESS BAR
 *********************************************/
.reveal .progress {
  position: absolute;
  display: none;
  height: 3px;
  width: 100%;
  bottom: 0;
  left: 0;
  z-index: 10;
  background-color: rgba(0, 0, 0, 0.2);
  color: #fff; }

.reveal .progress:after {
  content: '';
  display: block;
  position: absolute;
  height: 10px;
  width: 100%;
  top: -10px; }

.reveal .progress span {
  display: block;
  height: 100%;
  width: 0px;
  background-color: currentColor;
  transition: width 800ms cubic-bezier(0.26, 0.86, 0.44, 0.985); }

/*********************************************
 * SLIDE NUMBER
 *********************************************/
.reveal .slide-number {
  position: absolute;
  display: block;
  right: 8px;
  bottom: 8px;
  z-index: 31;
  font-family: Helvetica, sans-serif;
  font-size: 12px;
  line-height: 1;
  color: #fff;
  background-color: rgba(0, 0, 0, 0.4);
  padding: 5px; }

.reveal .slide-number a {
  color: currentColor; }

.reveal .slide-number-delimiter {
  margin: 0 3px; }

/*********************************************
 * SLIDES
 *********************************************/
.reveal {
  position: relative;
  width: 100%;
  height: 100%;
  overflow: hidden;
  -ms-touch-action: none;
      touch-action: none; }

@media only screen and (orientation: landscape) {
  .reveal.ua-iphone {
    position: fixed; } }

.reveal .slides {
  position: absolute;
  width: 100%;
  height: 100%;
  top: 0;
  right: 0;
  bottom: 0;
  left: 0;
  margin: auto;
  pointer-events: none;
  overflow: visible;
  z-index: 1;
  text-align: center;
  -webkit-perspective: 600px;
          perspective: 600px;
  -webkit-perspective-origin: 50% 40%;
          perspective-origin: 50% 40%; }

.reveal .slides > section {
  -ms-perspective: 600px; }

.reveal .slides > section,
.reveal .slides > section > section {
  display: none;
  position: absolute;
  width: 100%;
  padding: 20px 0px;
  pointer-events: auto;
  z-index: 10;
  -webkit-transform-style: flat;
          transform-style: flat;
  transition: -webkit-transform-origin 800ms cubic-bezier(0.26, 0.86, 0.44, 0.985), -webkit-transform 800ms cubic-bezier(0.26, 0.86, 0.44, 0.985), visibility 800ms cubic-bezier(0.26, 0.86, 0.44, 0.985), opacity 800ms cubic-bezier(0.26, 0.86, 0.44, 0.985);
  transition: transform-origin 800ms cubic-bezier(0.26, 0.86, 0.44, 0.985), transform 800ms cubic-bezier(0.26, 0.86, 0.44, 0.985), visibility 800ms cubic-bezier(0.26, 0.86, 0.44, 0.985), opacity 800ms cubic-bezier(0.26, 0.86, 0.44, 0.985); }

/* Global transition speed settings */
.reveal[data-transition-speed="fast"] .slides section {
  transition-duration: 400ms; }

.reveal[data-transition-speed="slow"] .slides section {
  transition-duration: 1200ms; }

/* Slide-specific transition speed overrides */
.reveal .slides section[data-transition-speed="fast"] {
  transition-duration: 400ms; }

.reveal .slides section[data-transition-speed="slow"] {
  transition-duration: 1200ms; }

.reveal .slides > section.stack {
  padding-top: 0;
  padding-bottom: 0;
  pointer-events: none; }

.reveal .slides > section.present,
.reveal .slides > section > section.present {
  display: block;
  z-index: 11;
  opacity: 1; }

.reveal .slides > section:empty,
.reveal .slides > section > section:empty,
.reveal .slides > section[data-background-interactive],
.reveal .slides > section > section[data-background-interactive] {
  pointer-events: none; }

.reveal.center,
.reveal.center .slides,
.reveal.center .slides section {
  min-height: 0 !important; }

/* Don't allow interaction with invisible slides */
.reveal .slides > section.future,
.reveal .slides > section > section.future,
.reveal .slides > section.past,
.reveal .slides > section > section.past {
  pointer-events: none; }

.reveal.overview .slides > section,
.reveal.overview .slides > section > section {
  pointer-events: auto; }

.reveal .slides > section.past,
.reveal .slides > section.future,
.reveal .slides > section > section.past,
.reveal .slides > section > section.future {
  opacity: 0; }

/*********************************************
 * Mixins for readability of transitions
 *********************************************/
/*********************************************
 * SLIDE TRANSITION
 * Aliased 'linear' for backwards compatibility
 *********************************************/
.reveal.slide section {
  -webkit-backface-visibility: hidden;
          backface-visibility: hidden; }

.reveal .slides > section[data-transition=slide].past,
.reveal .slides > section[data-transition~=slide-out].past,
.reveal.slide .slides > section:not([data-transition]).past {
  -webkit-transform: translate(-150%, 0);
          transform: translate(-150%, 0); }

.reveal .slides > section[data-transition=slide].future,
.reveal .slides > section[data-transition~=slide-in].future,
.reveal.slide .slides > section:not([data-transition]).future {
  -webkit-transform: translate(150%, 0);
          transform: translate(150%, 0); }

.reveal .slides > section > section[data-transition=slide].past,
.reveal .slides > section > section[data-transition~=slide-out].past,
.reveal.slide .slides > section > section:not([data-transition]).past {
  -webkit-transform: translate(0, -150%);
          transform: translate(0, -150%); }

.reveal .slides > section > section[data-transition=slide].future,
.reveal .slides > section > section[data-transition~=slide-in].future,
.reveal.slide .slides > section > section:not([data-transition]).future {
  -webkit-transform: translate(0, 150%);
          transform: translate(0, 150%); }

.reveal.linear section {
  -webkit-backface-visibility: hidden;
          backface-visibility: hidden; }

.reveal .slides > section[data-transition=linear].past,
.reveal .slides > section[data-transition~=linear-out].past,
.reveal.linear .slides > section:not([data-transition]).past {
  -webkit-transform: translate(-150%, 0);
          transform: translate(-150%, 0); }

.reveal .slides > section[data-transition=linear].future,
.reveal .slides > section[data-transition~=linear-in].future,
.reveal.linear .slides > section:not([data-transition]).future {
  -webkit-transform: translate(150%, 0);
          transform: translate(150%, 0); }

.reveal .slides > section > section[data-transition=linear].past,
.reveal .slides > section > section[data-transition~=linear-out].past,
.reveal.linear .slides > section > section:not([data-transition]).past {
  -webkit-transform: translate(0, -150%);
          transform: translate(0, -150%); }

.reveal .slides > section > section[data-transition=linear].future,
.reveal .slides > section > section[data-transition~=linear-in].future,
.reveal.linear .slides > section > section:not([data-transition]).future {
  -webkit-transform: translate(0, 150%);
          transform: translate(0, 150%); }

/*********************************************
 * CONVEX TRANSITION
 * Aliased 'default' for backwards compatibility
 *********************************************/
.reveal .slides section[data-transition=default].stack,
.reveal.default .slides section.stack {
  -webkit-transform-style: preserve-3d;
          transform-style: preserve-3d; }

.reveal .slides > section[data-transition=default].past,
.reveal .slides > section[data-transition~=default-out].past,
.reveal.default .slides > section:not([data-transition]).past {
  -webkit-transform: translate3d(-100%, 0, 0) rotateY(-90deg) translate3d(-100%, 0, 0);
          transform: translate3d(-100%, 0, 0) rotateY(-90deg) translate3d(-100%, 0, 0); }

.reveal .slides > section[data-transition=default].future,
.reveal .slides > section[data-transition~=default-in].future,
.reveal.default .slides > section:not([data-transition]).future {
  -webkit-transform: translate3d(100%, 0, 0) rotateY(90deg) translate3d(100%, 0, 0);
          transform: translate3d(100%, 0, 0) rotateY(90deg) translate3d(100%, 0, 0); }

.reveal .slides > section > section[data-transition=default].past,
.reveal .slides > section > section[data-transition~=default-out].past,
.reveal.default .slides > section > section:not([data-transition]).past {
  -webkit-transform: translate3d(0, -300px, 0) rotateX(70deg) translate3d(0, -300px, 0);
          transform: translate3d(0, -300px, 0) rotateX(70deg) translate3d(0, -300px, 0); }

.reveal .slides > section > section[data-transition=default].future,
.reveal .slides > section > section[data-transition~=default-in].future,
.reveal.default .slides > section > section:not([data-transition]).future {
  -webkit-transform: translate3d(0, 300px, 0) rotateX(-70deg) translate3d(0, 300px, 0);
          transform: translate3d(0, 300px, 0) rotateX(-70deg) translate3d(0, 300px, 0); }

.reveal .slides section[data-transition=convex].stack,
.reveal.convex .slides section.stack {
  -webkit-transform-style: preserve-3d;
          transform-style: preserve-3d; }

.reveal .slides > section[data-transition=convex].past,
.reveal .slides > section[data-transition~=convex-out].past,
.reveal.convex .slides > section:not([data-transition]).past {
  -webkit-transform: translate3d(-100%, 0, 0) rotateY(-90deg) translate3d(-100%, 0, 0);
          transform: translate3d(-100%, 0, 0) rotateY(-90deg) translate3d(-100%, 0, 0); }

.reveal .slides > section[data-transition=convex].future,
.reveal .slides > section[data-transition~=convex-in].future,
.reveal.convex .slides > section:not([data-transition]).future {
  -webkit-transform: translate3d(100%, 0, 0) rotateY(90deg) translate3d(100%, 0, 0);
          transform: translate3d(100%, 0, 0) rotateY(90deg) translate3d(100%, 0, 0); }

.reveal .slides > section > section[data-transition=convex].past,
.reveal .slides > section > section[data-transition~=convex-out].past,
.reveal.convex .slides > section > section:not([data-transition]).past {
  -webkit-transform: translate3d(0, -300px, 0) rotateX(70deg) translate3d(0, -300px, 0);
          transform: translate3d(0, -300px, 0) rotateX(70deg) translate3d(0, -300px, 0); }

.reveal .slides > section > section[data-transition=convex].future,
.reveal .slides > section > section[data-transition~=convex-in].future,
.reveal.convex .slides > section > section:not([data-transition]).future {
  -webkit-transform: translate3d(0, 300px, 0) rotateX(-70deg) translate3d(0, 300px, 0);
          transform: translate3d(0, 300px, 0) rotateX(-70deg) translate3d(0, 300px, 0); }

/*********************************************
 * CONCAVE TRANSITION
 *********************************************/
.reveal .slides section[data-transition=concave].stack,
.reveal.concave .slides section.stack {
  -webkit-transform-style: preserve-3d;
          transform-style: preserve-3d; }

.reveal .slides > section[data-transition=concave].past,
.reveal .slides > section[data-transition~=concave-out].past,
.reveal.concave .slides > section:not([data-transition]).past {
  -webkit-transform: translate3d(-100%, 0, 0) rotateY(90deg) translate3d(-100%, 0, 0);
          transform: translate3d(-100%, 0, 0) rotateY(90deg) translate3d(-100%, 0, 0); }

.reveal .slides > section[data-transition=concave].future,
.reveal .slides > section[data-transition~=concave-in].future,
.reveal.concave .slides > section:not([data-transition]).future {
  -webkit-transform: translate3d(100%, 0, 0) rotateY(-90deg) translate3d(100%, 0, 0);
          transform: translate3d(100%, 0, 0) rotateY(-90deg) translate3d(100%, 0, 0); }

.reveal .slides > section > section[data-transition=concave].past,
.reveal .slides > section > section[data-transition~=concave-out].past,
.reveal.concave .slides > section > section:not([data-transition]).past {
  -webkit-transform: translate3d(0, -80%, 0) rotateX(-70deg) translate3d(0, -80%, 0);
          transform: translate3d(0, -80%, 0) rotateX(-70deg) translate3d(0, -80%, 0); }

.reveal .slides > section > section[data-transition=concave].future,
.reveal .slides > section > section[data-transition~=concave-in].future,
.reveal.concave .slides > section > section:not([data-transition]).future {
  -webkit-transform: translate3d(0, 80%, 0) rotateX(70deg) translate3d(0, 80%, 0);
          transform: translate3d(0, 80%, 0) rotateX(70deg) translate3d(0, 80%, 0); }

/*********************************************
 * ZOOM TRANSITION
 *********************************************/
.reveal .slides section[data-transition=zoom],
.reveal.zoom .slides section:not([data-transition]) {
  transition-timing-function: ease; }

.reveal .slides > section[data-transition=zoom].past,
.reveal .slides > section[data-transition~=zoom-out].past,
.reveal.zoom .slides > section:not([data-transition]).past {
  visibility: hidden;
  -webkit-transform: scale(16);
          transform: scale(16); }

.reveal .slides > section[data-transition=zoom].future,
.reveal .slides > section[data-transition~=zoom-in].future,
.reveal.zoom .slides > section:not([data-transition]).future {
  visibility: hidden;
  -webkit-transform: scale(0.2);
          transform: scale(0.2); }

.reveal .slides > section > section[data-transition=zoom].past,
.reveal .slides > section > section[data-transition~=zoom-out].past,
.reveal.zoom .slides > section > section:not([data-transition]).past {
  -webkit-transform: translate(0, -150%);
          transform: translate(0, -150%); }

.reveal .slides > section > section[data-transition=zoom].future,
.reveal .slides > section > section[data-transition~=zoom-in].future,
.reveal.zoom .slides > section > section:not([data-transition]).future {
  -webkit-transform: translate(0, 150%);
          transform: translate(0, 150%); }

/*********************************************
 * CUBE TRANSITION
 *
 * WARNING:
 * this is deprecated and will be removed in a
 * future version.
 *********************************************/
.reveal.cube .slides {
  -webkit-perspective: 1300px;
          perspective: 1300px; }

.reveal.cube .slides section {
  padding: 30px;
  min-height: 700px;
  -webkit-backface-visibility: hidden;
          backface-visibility: hidden;
  box-sizing: border-box;
  -webkit-transform-style: preserve-3d;
          transform-style: preserve-3d; }

.reveal.center.cube .slides section {
  min-height: 0; }

.reveal.cube .slides section:not(.stack):before {
  content: '';
  position: absolute;
  display: block;
  width: 100%;
  height: 100%;
  left: 0;
  top: 0;
  background: rgba(0, 0, 0, 0.1);
  border-radius: 4px;
  -webkit-transform: translateZ(-20px);
          transform: translateZ(-20px); }

.reveal.cube .slides section:not(.stack):after {
  content: '';
  position: absolute;
  display: block;
  width: 90%;
  height: 30px;
  left: 5%;
  bottom: 0;
  background: none;
  z-index: 1;
  border-radius: 4px;
  box-shadow: 0px 95px 25px rgba(0, 0, 0, 0.2);
  -webkit-transform: translateZ(-90px) rotateX(65deg);
          transform: translateZ(-90px) rotateX(65deg); }

.reveal.cube .slides > section.stack {
  padding: 0;
  background: none; }

.reveal.cube .slides > section.past {
  -webkit-transform-origin: 100% 0%;
          transform-origin: 100% 0%;
  -webkit-transform: translate3d(-100%, 0, 0) rotateY(-90deg);
          transform: translate3d(-100%, 0, 0) rotateY(-90deg); }

.reveal.cube .slides > section.future {
  -webkit-transform-origin: 0% 0%;
          transform-origin: 0% 0%;
  -webkit-transform: translate3d(100%, 0, 0) rotateY(90deg);
          transform: translate3d(100%, 0, 0) rotateY(90deg); }

.reveal.cube .slides > section > section.past {
  -webkit-transform-origin: 0% 100%;
          transform-origin: 0% 100%;
  -webkit-transform: translate3d(0, -100%, 0) rotateX(90deg);
          transform: translate3d(0, -100%, 0) rotateX(90deg); }

.reveal.cube .slides > section > section.future {
  -webkit-transform-origin: 0% 0%;
          transform-origin: 0% 0%;
  -webkit-transform: translate3d(0, 100%, 0) rotateX(-90deg);
          transform: translate3d(0, 100%, 0) rotateX(-90deg); }

/*********************************************
 * PAGE TRANSITION
 *
 * WARNING:
 * this is deprecated and will be removed in a
 * future version.
 *********************************************/
.reveal.page .slides {
  -webkit-perspective-origin: 0% 50%;
          perspective-origin: 0% 50%;
  -webkit-perspective: 3000px;
          perspective: 3000px; }

.reveal.page .slides section {
  padding: 30px;
  min-height: 700px;
  box-sizing: border-box;
  -webkit-transform-style: preserve-3d;
          transform-style: preserve-3d; }

.reveal.page .slides section.past {
  z-index: 12; }

.reveal.page .slides section:not(.stack):before {
  content: '';
  position: absolute;
  display: block;
  width: 100%;
  height: 100%;
  left: 0;
  top: 0;
  background: rgba(0, 0, 0, 0.1);
  -webkit-transform: translateZ(-20px);
          transform: translateZ(-20px); }

.reveal.page .slides section:not(.stack):after {
  content: '';
  position: absolute;
  display: block;
  width: 90%;
  height: 30px;
  left: 5%;
  bottom: 0;
  background: none;
  z-index: 1;
  border-radius: 4px;
  box-shadow: 0px 95px 25px rgba(0, 0, 0, 0.2);
  -webkit-transform: translateZ(-90px) rotateX(65deg); }

.reveal.page .slides > section.stack {
  padding: 0;
  background: none; }

.reveal.page .slides > section.past {
  -webkit-transform-origin: 0% 0%;
          transform-origin: 0% 0%;
  -webkit-transform: translate3d(-40%, 0, 0) rotateY(-80deg);
          transform: translate3d(-40%, 0, 0) rotateY(-80deg); }

.reveal.page .slides > section.future {
  -webkit-transform-origin: 100% 0%;
          transform-origin: 100% 0%;
  -webkit-transform: translate3d(0, 0, 0);
          transform: translate3d(0, 0, 0); }

.reveal.page .slides > section > section.past {
  -webkit-transform-origin: 0% 0%;
          transform-origin: 0% 0%;
  -webkit-transform: translate3d(0, -40%, 0) rotateX(80deg);
          transform: translate3d(0, -40%, 0) rotateX(80deg); }

.reveal.page .slides > section > section.future {
  -webkit-transform-origin: 0% 100%;
          transform-origin: 0% 100%;
  -webkit-transform: translate3d(0, 0, 0);
          transform: translate3d(0, 0, 0); }

/*********************************************
 * FADE TRANSITION
 *********************************************/
.reveal .slides section[data-transition=fade],
.reveal.fade .slides section:not([data-transition]),
.reveal.fade .slides > section > section:not([data-transition]) {
  -webkit-transform: none;
          transform: none;
  transition: opacity 0.5s; }

.reveal.fade.overview .slides section,
.reveal.fade.overview .slides > section > section {
  transition: none; }

/*********************************************
 * NO TRANSITION
 *********************************************/
.reveal .slides section[data-transition=none],
.reveal.none .slides section:not([data-transition]) {
  -webkit-transform: none;
          transform: none;
  transition: none; }

/*********************************************
 * PAUSED MODE
 *********************************************/
.reveal .pause-overlay {
  position: absolute;
  top: 0;
  left: 0;
  width: 100%;
  height: 100%;
  background: black;
  visibility: hidden;
  opacity: 0;
  z-index: 100;
  transition: all 1s ease; }

.reveal .pause-overlay .resume-button {
  position: absolute;
  bottom: 20px;
  right: 20px;
  color: #ccc;
  border-radius: 2px;
  padding: 6px 14px;
  border: 2px solid #ccc;
  font-size: 16px;
  background: transparent;
  cursor: pointer; }
  .reveal .pause-overlay .resume-button:hover {
    color: #fff;
    border-color: #fff; }

.reveal.paused .pause-overlay {
  visibility: visible;
  opacity: 1; }

/*********************************************
 * FALLBACK
 *********************************************/
.no-transforms {
  overflow-y: auto; }

.no-transforms .reveal .slides {
  position: relative;
  width: 80%;
  height: auto !important;
  top: 0;
  left: 50%;
  margin: 0;
  text-align: center; }

.no-transforms .reveal .controls,
.no-transforms .reveal .progress {
  display: none !important; }

.no-transforms .reveal .slides section {
  display: block !important;
  opacity: 1 !important;
  position: relative !important;
  height: auto;
  min-height: 0;
  top: 0;
  left: -50%;
  margin: 70px 0;
  -webkit-transform: none;
          transform: none; }

.no-transforms .reveal .slides section section {
  left: 0; }

.reveal .no-transition,
.reveal .no-transition * {
  transition: none !important; }

/*********************************************
 * PER-SLIDE BACKGROUNDS
 *********************************************/
.reveal .backgrounds {
  position: absolute;
  width: 100%;
  height: 100%;
  top: 0;
  left: 0;
  -webkit-perspective: 600px;
          perspective: 600px; }

.reveal .slide-background {
  display: none;
  position: absolute;
  width: 100%;
  height: 100%;
  opacity: 0;
  visibility: hidden;
  overflow: hidden;
  background-color: transparent;
  transition: all 800ms cubic-bezier(0.26, 0.86, 0.44, 0.985); }

.reveal .slide-background-content {
  position: absolute;
  width: 100%;
  height: 100%;
  background-position: 50% 50%;
  background-repeat: no-repeat;
  background-size: cover; }

.reveal .slide-background.stack {
  display: block; }

.reveal .slide-background.present {
  opacity: 1;
  visibility: visible;
  z-index: 2; }

.print-pdf .reveal .slide-background {
  opacity: 1 !important;
  visibility: visible !important; }

/* Video backgrounds */
.reveal .slide-background video {
  position: absolute;
  width: 100%;
  height: 100%;
  max-width: none;
  max-height: none;
  top: 0;
  left: 0;
  -o-object-fit: cover;
     object-fit: cover; }

.reveal .slide-background[data-background-size="contain"] video {
  -o-object-fit: contain;
     object-fit: contain; }

/* Immediate transition style */
.reveal[data-background-transition=none] > .backgrounds .slide-background,
.reveal > .backgrounds .slide-background[data-background-transition=none] {
  transition: none; }

/* Slide */
.reveal[data-background-transition=slide] > .backgrounds .slide-background,
.reveal > .backgrounds .slide-background[data-background-transition=slide] {
  opacity: 1;
  -webkit-backface-visibility: hidden;
          backface-visibility: hidden; }

.reveal[data-background-transition=slide] > .backgrounds .slide-background.past,
.reveal > .backgrounds .slide-background.past[data-background-transition=slide] {
  -webkit-transform: translate(-100%, 0);
          transform: translate(-100%, 0); }

.reveal[data-background-transition=slide] > .backgrounds .slide-background.future,
.reveal > .backgrounds .slide-background.future[data-background-transition=slide] {
  -webkit-transform: translate(100%, 0);
          transform: translate(100%, 0); }

.reveal[data-background-transition=slide] > .backgrounds .slide-background > .slide-background.past,
.reveal > .backgrounds .slide-background > .slide-background.past[data-background-transition=slide] {
  -webkit-transform: translate(0, -100%);
          transform: translate(0, -100%); }

.reveal[data-background-transition=slide] > .backgrounds .slide-background > .slide-background.future,
.reveal > .backgrounds .slide-background > .slide-background.future[data-background-transition=slide] {
  -webkit-transform: translate(0, 100%);
          transform: translate(0, 100%); }

/* Convex */
.reveal[data-background-transition=convex] > .backgrounds .slide-background.past,
.reveal > .backgrounds .slide-background.past[data-background-transition=convex] {
  opacity: 0;
  -webkit-transform: translate3d(-100%, 0, 0) rotateY(-90deg) translate3d(-100%, 0, 0);
          transform: translate3d(-100%, 0, 0) rotateY(-90deg) translate3d(-100%, 0, 0); }

.reveal[data-background-transition=convex] > .backgrounds .slide-background.future,
.reveal > .backgrounds .slide-background.future[data-background-transition=convex] {
  opacity: 0;
  -webkit-transform: translate3d(100%, 0, 0) rotateY(90deg) translate3d(100%, 0, 0);
          transform: translate3d(100%, 0, 0) rotateY(90deg) translate3d(100%, 0, 0); }

.reveal[data-background-transition=convex] > .backgrounds .slide-background > .slide-background.past,
.reveal > .backgrounds .slide-background > .slide-background.past[data-background-transition=convex] {
  opacity: 0;
  -webkit-transform: translate3d(0, -100%, 0) rotateX(90deg) translate3d(0, -100%, 0);
          transform: translate3d(0, -100%, 0) rotateX(90deg) translate3d(0, -100%, 0); }

.reveal[data-background-transition=convex] > .backgrounds .slide-background > .slide-background.future,
.reveal > .backgrounds .slide-background > .slide-background.future[data-background-transition=convex] {
  opacity: 0;
  -webkit-transform: translate3d(0, 100%, 0) rotateX(-90deg) translate3d(0, 100%, 0);
          transform: translate3d(0, 100%, 0) rotateX(-90deg) translate3d(0, 100%, 0); }

/* Concave */
.reveal[data-background-transition=concave] > .backgrounds .slide-background.past,
.reveal > .backgrounds .slide-background.past[data-background-transition=concave] {
  opacity: 0;
  -webkit-transform: translate3d(-100%, 0, 0) rotateY(90deg) translate3d(-100%, 0, 0);
          transform: translate3d(-100%, 0, 0) rotateY(90deg) translate3d(-100%, 0, 0); }

.reveal[data-background-transition=concave] > .backgrounds .slide-background.future,
.reveal > .backgrounds .slide-background.future[data-background-transition=concave] {
  opacity: 0;
  -webkit-transform: translate3d(100%, 0, 0) rotateY(-90deg) translate3d(100%, 0, 0);
          transform: translate3d(100%, 0, 0) rotateY(-90deg) translate3d(100%, 0, 0); }

.reveal[data-background-transition=concave] > .backgrounds .slide-background > .slide-background.past,
.reveal > .backgrounds .slide-background > .slide-background.past[data-background-transition=concave] {
  opacity: 0;
  -webkit-transform: translate3d(0, -100%, 0) rotateX(-90deg) translate3d(0, -100%, 0);
          transform: translate3d(0, -100%, 0) rotateX(-90deg) translate3d(0, -100%, 0); }

.reveal[data-background-transition=concave] > .backgrounds .slide-background > .slide-background.future,
.reveal > .backgrounds .slide-background > .slide-background.future[data-background-transition=concave] {
  opacity: 0;
  -webkit-transform: translate3d(0, 100%, 0) rotateX(90deg) translate3d(0, 100%, 0);
          transform: translate3d(0, 100%, 0) rotateX(90deg) translate3d(0, 100%, 0); }

/* Zoom */
.reveal[data-background-transition=zoom] > .backgrounds .slide-background,
.reveal > .backgrounds .slide-background[data-background-transition=zoom] {
  transition-timing-function: ease; }

.reveal[data-background-transition=zoom] > .backgrounds .slide-background.past,
.reveal > .backgrounds .slide-background.past[data-background-transition=zoom] {
  opacity: 0;
  visibility: hidden;
  -webkit-transform: scale(16);
          transform: scale(16); }

.reveal[data-background-transition=zoom] > .backgrounds .slide-background.future,
.reveal > .backgrounds .slide-background.future[data-background-transition=zoom] {
  opacity: 0;
  visibility: hidden;
  -webkit-transform: scale(0.2);
          transform: scale(0.2); }

.reveal[data-background-transition=zoom] > .backgrounds .slide-background > .slide-background.past,
.reveal > .backgrounds .slide-background > .slide-background.past[data-background-transition=zoom] {
  opacity: 0;
  visibility: hidden;
  -webkit-transform: scale(16);
          transform: scale(16); }

.reveal[data-background-transition=zoom] > .backgrounds .slide-background > .slide-background.future,
.reveal > .backgrounds .slide-background > .slide-background.future[data-background-transition=zoom] {
  opacity: 0;
  visibility: hidden;
  -webkit-transform: scale(0.2);
          transform: scale(0.2); }

/* Global transition speed settings */
.reveal[data-transition-speed="fast"] > .backgrounds .slide-background {
  transition-duration: 400ms; }

.reveal[data-transition-speed="slow"] > .backgrounds .slide-background {
  transition-duration: 1200ms; }

/*********************************************
 * OVERVIEW
 *********************************************/
.reveal.overview {
  -webkit-perspective-origin: 50% 50%;
          perspective-origin: 50% 50%;
  -webkit-perspective: 700px;
          perspective: 700px; }
  .reveal.overview .slides {
    -moz-transform-style: preserve-3d; }
  .reveal.overview .slides section {
    height: 100%;
    top: 0 !important;
    opacity: 1 !important;
    overflow: hidden;
    visibility: visible !important;
    cursor: pointer;
    box-sizing: border-box; }
  .reveal.overview .slides section:hover,
  .reveal.overview .slides section.present {
    outline: 10px solid rgba(150, 150, 150, 0.4);
    outline-offset: 10px; }
  .reveal.overview .slides section .fragment {
    opacity: 1;
    transition: none; }
  .reveal.overview .slides section:after,
  .reveal.overview .slides section:before {
    display: none !important; }
  .reveal.overview .slides > section.stack {
    padding: 0;
    top: 0 !important;
    background: none;
    outline: none;
    overflow: visible; }
  .reveal.overview .backgrounds {
    -webkit-perspective: inherit;
            perspective: inherit;
    -moz-transform-style: preserve-3d; }
  .reveal.overview .backgrounds .slide-background {
    opacity: 1;
    visibility: visible;
    outline: 10px solid rgba(150, 150, 150, 0.1);
    outline-offset: 10px; }
  .reveal.overview .backgrounds .slide-background.stack {
    overflow: visible; }

.reveal.overview .slides section,
.reveal.overview-deactivating .slides section {
  transition: none; }

.reveal.overview .backgrounds .slide-background,
.reveal.overview-deactivating .backgrounds .slide-background {
  transition: none; }

/*********************************************
 * RTL SUPPORT
 *********************************************/
.reveal.rtl .slides,
.reveal.rtl .slides h1,
.reveal.rtl .slides h2,
.reveal.rtl .slides h3,
.reveal.rtl .slides h4,
.reveal.rtl .slides h5,
.reveal.rtl .slides h6 {
  direction: rtl;
  font-family: sans-serif; }

.reveal.rtl pre,
.reveal.rtl code {
  direction: ltr; }

.reveal.rtl ol,
.reveal.rtl ul {
  text-align: right; }

.reveal.rtl .progress span {
  float: right; }

/*********************************************
 * PARALLAX BACKGROUND
 *********************************************/
.reveal.has-parallax-background .backgrounds {
  transition: all 0.8s ease; }

/* Global transition speed settings */
.reveal.has-parallax-background[data-transition-speed="fast"] .backgrounds {
  transition-duration: 400ms; }

.reveal.has-parallax-background[data-transition-speed="slow"] .backgrounds {
  transition-duration: 1200ms; }

/*********************************************
 * LINK PREVIEW OVERLAY
 *********************************************/
.reveal .overlay {
  position: absolute;
  top: 0;
  left: 0;
  width: 100%;
  height: 100%;
  z-index: 1000;
  background: rgba(0, 0, 0, 0.9);
  opacity: 0;
  visibility: hidden;
  transition: all 0.3s ease; }

.reveal .overlay.visible {
  opacity: 1;
  visibility: visible; }

.reveal .overlay .spinner {
  position: absolute;
  display: block;
  top: 50%;
  left: 50%;
  width: 32px;
  height: 32px;
  margin: -16px 0 0 -16px;
  z-index: 10;
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    <h1 style="width: 50%">
        &#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;<br> 
        <span style="font-size: 0.6em">&#x8C31;&#x56FE;&#x7406;&#x8BBA;</span>
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    <h2>&#x5F20;&#x817E;</h2>
    <h3>tengzhang@hust.edu.cn</h3>
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</section><section><section data-notes lineno="25" class="slide " data-line="25" data-h="1" data-v="0">
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        <h5 class="title">Laplace&#x77E9;&#x9635;</h5>
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<p>&#x5B9A;&#x4E49;&#xFF1A;<span class="mathjax-exps">$\Lv = \Dv - \Av$</span></p>
<ul>
<li><span class="mathjax-exps">$\Lv$</span>&#x534A;&#x6B63;&#x5B9A;&#xFF1A;&#x5BF9;<span class="mathjax-exps">$\forall \xv \in \Rbb^{|\Vcal|}$</span></li>
</ul>
<div>
    $$
        \begin{align*}
            \xv^\top &amp; \Lv \xv = \sum_{u \in \Vcal} \sum_{v \in \Vcal} [\xv]_u [\Dv]_{uv} [\xv]_v - \sum_{u \in \Vcal} \sum_{v \in \Vcal} [\xv]_u [\Av]_{uv} [\xv]_v \\
                        &amp; = \sum_{u \in \Vcal} [\xv]_u^2 d_u - \sum_{u \in \Vcal} \sum_{v \in \Vcal} [\xv]_u [\Av]_{uv} [\xv]_v                          \\
                        &amp; = \frac{1}{2} \left( \sum_{u \in \Vcal} \sum_{v \in \Vcal} ([\xv]_u^2 [\Av]_{uv} + [\xv]_v^2 [\Av]_{vu}) - 2 \sum_{u \in \Vcal} \sum_{v \in \Vcal} [\xv]_u [\Av]_{uv} [\xv]_v \right) \\
                        &amp; = \frac{1}{2} \sum_{u \in \Vcal} \sum_{v \in \Vcal} [\Av]_{uv} ([\xv]_u - [\xv]_v)^2 \geq 0
        \end{align*}
    $$
</div>
<ul>
<li><span class="mathjax-exps">$\Lv \ev = \Dv \ev - \Av \ev = \zerov = 0 \ev$</span>&#xFF0C;&#x6545;<span class="mathjax-exps">$0 = \lambda_{|\Vcal|} \leq \cdots \leq \lambda_1$</span></li>
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        <h6 class="bottom_center">&#x8C31;&#x56FE;&#x7406;&#x8BBA;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
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</section><section vertical="true" data-notes lineno="60" class="slide " data-line="60" data-h="1" data-v="1">
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<div>
    $$
        \begin{align*}
            \xv^\top \Lv \xv = \frac{1}{2} \sum_{u \in \Vcal} \sum_{v \in \Vcal} [\Av]_{uv} ([\xv]_u - [\xv]_v)^2 = \frac{1}{2} \sum_{(u,v) \in \Ecal} ([\xv]_u - [\xv]_v)^2
        \end{align*}
    $$
</div>
<p>Laplace &#x77E9;&#x9635;&#x6700;&#x5C0F;&#x7279;&#x5F81;&#x503C;<span class="mathjax-exps">$0$</span>&#x7684;&#x51E0;&#x4F55;&#x91CD;&#x6570;&#x7B49;&#x4E8E;&#x56FE;&#x7684;&#x8FDE;&#x901A;&#x5206;&#x652F;&#x6570;</p>
<p>&#x8BC1;&#x660E;&#xFF1A;&#x8BBE;<span class="mathjax-exps">$\xv$</span>&#x662F;<span class="mathjax-exps">$0$</span>&#x5BF9;&#x5E94;&#x7684;&#x4EFB;&#x4E00;&#x7279;&#x5F81;&#x5411;&#x91CF;&#xFF0C;&#x4E8E;&#x662F;&#x6709;<span class="mathjax-exps">$\xv^\top \Lv \xv = \xv^\top 0 \xv = 0$</span>&#xFF0C;&#x4ECE;&#x800C;<span class="mathjax-exps">$(u,v) \in \Ecal \Rightarrow [\xv]_u = [\xv]_v$</span>&#xFF0C;&#x8FDB;&#x4E00;&#x6B65;<span class="mathjax-exps">$\xv$</span>&#x5728;&#x4EFB;&#x4E00;&#x8FDE;&#x901A;&#x5206;&#x652F;&#x4E0A;&#x7684;&#x503C;&#x76F8;&#x7B49;</p>
<ul>
<li>&#x5168;&#x8FDE;&#x901A;&#x56FE;&#x53EA;&#x6709;&#x4E00;&#x4E2A;&#x8FDE;&#x901A;&#x5206;&#x652F;&#xFF0C;&#x6B64;&#x65F6;<span class="mathjax-exps">$\xv = c \ev$</span>&#xFF0C;&#x5373;<span class="mathjax-exps">$0$</span>&#x7684;&#x51E0;&#x4F55;&#x91CD;&#x6570;&#x4E3A;<span class="mathjax-exps">$1$</span></li>
<li>&#x8BBE;&#x56FE;&#x6709;<span class="mathjax-exps">$K$</span>&#x4E2A;&#x8FDE;&#x901A;&#x5206;&#x652F;<span class="mathjax-exps">$\Vcal = \Vcal_1 \times \cdots \times \Vcal_K$</span>&#xFF0C;&#x5C06;&#x70B9;&#x6309;&#x8FDE;&#x901A;&#x5206;&#x652F;&#x6392;&#x5E8F;&#xFF0C;&#x5219;&#x90BB;&#x63A5;&#x77E9;&#x9635;&#x4E3A;&#x5206;&#x5757;&#x5BF9;&#x89D2;&#x9635;<span class="mathjax-exps">$\Av = \diag\{ \Av_1, \ldots, \Av_K \}$</span>&#xFF0C;&#x53C8;<span class="mathjax-exps">$\Dv$</span>&#x662F;&#x5BF9;&#x89D2;&#x9635;&#xFF0C;&#x6545; Laplace &#x77E9;&#x9635;&#x4E5F;&#x4E3A;&#x5206;&#x5757;&#x5BF9;&#x89D2;&#x9635;<span class="mathjax-exps">$\Lv = \diag\{ \Lv_1, \ldots, \Lv_K \}$</span>&#xFF0C;&#x5176;&#x4E2D;<span class="mathjax-exps">$\Lv_i$</span>&#x662F;&#x7B2C;<span class="mathjax-exps">$i$</span>&#x4E2A;&#x8FDE;&#x901A;&#x5206;&#x652F;&#x7684; Laplace &#x77E9;&#x9635;&#xFF0C;&#x5176;&#x6700;&#x5C0F;&#x7279;&#x5F81;&#x503C;<span class="mathjax-exps">$0$</span>&#x5BF9;&#x5E94;&#x7684;&#x7279;&#x5F81;&#x5411;&#x91CF;<span class="mathjax-exps">$\ev \in \Rbb^{|\Vcal_i|}$</span>&#xFF0C;&#x5C06;&#x5176;&#x8865;&#x96F6;&#x5230;<span class="mathjax-exps">$|\Vcal|$</span>&#x7EF4;&#x5F97;&#x5230;<span class="mathjax-exps">$\ev_{\Vcal_i}$</span>&#xFF0C;&#x663E;&#x7136;<span class="mathjax-exps">$\ev_{\Vcal_1}, \ldots, \ev_{\Vcal_K}$</span>&#x662F;<span class="mathjax-exps">$\Lv$</span>&#x7684;&#x5BF9;&#x5E94;&#x4E8E;<span class="mathjax-exps">$0$</span>&#x7684;<span class="mathjax-exps">$K$</span>&#x4E2A;&#x7EBF;&#x6027;&#x65E0;&#x5173;&#x7684;&#x7279;&#x5F81;&#x5411;&#x91CF;</li>
</ul>
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        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x8C31;&#x56FE;&#x7406;&#x8BBA;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
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</section></section><section><section data-notes lineno="93" class="slide " data-line="93" data-h="2" data-v="0">
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        <h5 class="title">&#x5F52;&#x4E00;&#x5316;Laplace&#x77E9;&#x9635;</h5>
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    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
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<p>&#x5BF9;&#x79F0; Laplace &#x77E9;&#x9635;&#xFF1A;<span class="mathjax-exps">$\Lv_{\text{sym}} = \Dv^{-1/2} \Lv \Dv^{-1/2} = \Iv - \Dv^{-1/2} \Av \Dv^{-1/2}$</span></p>
<p>&#x968F;&#x673A;&#x6E38;&#x8D70; Laplace &#x77E9;&#x9635;&#xFF1A;<span class="mathjax-exps">$\Lv_{\text{rm}} = \Dv^{-1} \Lv = \Iv - \Dv^{-1} \Av$</span>&#xFF0C;&#x7B2C;&#x4E8C;&#x9879;&#x5C31;&#x662F;&#x968F;&#x673A;&#x6E38;&#x8D70;&#x7684;&#x8F6C;&#x79FB;&#x77E9;&#x9635;&#xFF0C;&#x7531;&#x6B64;&#x5F97;&#x540D;</p>
<div>
    $$
        \begin{align*}
            \xv^\top \Lv_{\text{sym}} \xv &amp; = \sum_{u \in \Vcal} [\xv]_u^2 - \sum_{u \in \Vcal} \sum_{v \in \Vcal} [\xv]_u [\Av]_{uv} [\xv]_v / \sqrt{d_u d_v} \\
                        &amp; = \sum_{u \in \Vcal} \sum_{v \in \Vcal} [\xv]_u^2 [\Av]_{uv} / d_u - \sum_{u \in \Vcal} \sum_{v \in \Vcal} [\xv]_u [\Av]_{uv} [\xv]_v / \sqrt{d_u d_v}                          \\
                        &amp; = \frac{1}{2} \sum_{u \in \Vcal} \sum_{v \in \Vcal} [\Av]_{vu} \left( [\xv]_u^2 / d_u + [\xv]_v^2 / d_v - 2 [\xv]_u [\xv]_v / \sqrt{d_u d_v} \right) \\
                        &amp; = \frac{1}{2} \sum_{u \in \Vcal} \sum_{v \in \Vcal} [\Av]_{uv} \left( \frac{[\xv]_u}{\sqrt{d_u}} - \frac{[\xv]_v}{\sqrt{d_v}} \right)^2
        \end{align*}
    $$
</div>
<p>&#x5373;<span class="mathjax-exps">$\Lv_{\text{sym}}$</span>&#x534A;&#x6B63;&#x5B9A;&#xFF0C;&#x6709;<span class="mathjax-exps">$|\Vcal|$</span>&#x4E2A;&#x975E;&#x8D1F;&#x7279;&#x5F81;&#x503C;&#xFF1A;<span class="mathjax-exps">$0 \leq \lambda_{|\Vcal|} \leq \cdots \leq \lambda_1$</span></p>
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        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x8C31;&#x56FE;&#x7406;&#x8BBA;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section><section vertical="true" data-notes lineno="128" class="slide " data-line="128" data-h="2" data-v="1">
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<p>&#x5BF9;&#x79F0; Laplace &#x77E9;&#x9635;&#xFF1A;<span class="mathjax-exps">$\Lv_{\text{sym}} = \Dv^{-1/2} \Lv \Dv^{-1/2} = \Iv - \Dv^{-1/2} \Av \Dv^{-1/2}$</span></p>
<p>&#x968F;&#x673A;&#x6E38;&#x8D70; Laplace &#x77E9;&#x9635;&#xFF1A;<span class="mathjax-exps">$\Lv_{\text{rm}} = \Dv^{-1} \Lv = \Iv - \Dv^{-1} \Av$</span></p>
<br>
<p><span class="mathjax-exps">$(\lambda, \vv)$</span>&#x662F;<span class="mathjax-exps">$\Lv_{\text{rm}}$</span>&#x7684;&#x7279;&#x5F81;&#x503C;/&#x5411;&#x91CF;&#x5F53;&#x4E14;&#x4EC5;&#x5F53;<span class="mathjax-exps">$\Lv \vv = \lambda \Dv \vv$</span></p>
<div>
    $$
        \begin{align*}
            \lambda \Dv \vv = \Dv \lambda \vv = \Dv \Lv_{\text{rm}} \vv = \Dv \Dv^{-1} \Lv \vv = \Lv \vv
        \end{align*}
    $$
</div>
<br>
<p><span class="mathjax-exps">$(\lambda, \vv)$</span>&#x662F;<span class="mathjax-exps">$\Lv_{\text{rm}}$</span>&#x7684;&#x7279;&#x5F81;&#x503C;/&#x5411;&#x91CF;&#x5F53;&#x4E14;&#x4EC5;&#x5F53;<span class="mathjax-exps">$(\lambda, \Dv^{1/2} \vv)$</span>&#x662F;<span class="mathjax-exps">$\Lv_{\text{sym}}$</span>&#x7684;&#x7279;&#x5F81;&#x503C;/&#x5411;&#x91CF;</p>
<div>
    $$
        \begin{align*}
            \lambda \vv &amp; = \Lv_{\text{rm}} \vv = \Dv^{-1} \Lv \vv = \Dv^{-1/2} \Dv^{-1/2} \Lv \Dv^{-1/2} \Dv^{1/2} \vv \\
            &amp; = \Dv^{-1/2} \Lv_{\text{sym}} \Dv^{1/2} \vv \\
            &amp; \Longleftrightarrow \lambda \Dv^{1/2} \vv = \Lv_{\text{sym}} \Dv^{1/2} \vv
        \end{align*}
    $$
</div>
<p>&#x6545;<span class="mathjax-exps">$\Lv_{\text{rm}}$</span>&#x4E5F;&#x534A;&#x6B63;&#x5B9A;&#xFF0C;&#x6709;<span class="mathjax-exps">$|\Vcal|$</span>&#x4E2A;&#x975E;&#x8D1F;&#x7279;&#x5F81;&#x503C;&#xFF1A;<span class="mathjax-exps">$0 \leq \lambda_{|\Vcal|} \leq \cdots \leq \lambda_1$</span></p>
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        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
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        <h5 class="title">&#x5F52;&#x4E00;&#x5316;Laplace&#x77E9;&#x9635;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>&#x5BF9;&#x79F0; Laplace &#x77E9;&#x9635;&#xFF1A;<span class="mathjax-exps">$\Lv_{\text{sym}} = \Dv^{-1/2} \Lv \Dv^{-1/2} = \Iv - \Dv^{-1/2} \Av \Dv^{-1/2}$</span></p>
<p>&#x968F;&#x673A;&#x6E38;&#x8D70; Laplace &#x77E9;&#x9635;&#xFF1A;<span class="mathjax-exps">$\Lv_{\text{rm}} = \Dv^{-1} \Lv = \Iv - \Dv^{-1} \Av$</span></p>
<br>
<p><span class="mathjax-exps">$(0, \ev)$</span>&#x662F;<span class="mathjax-exps">$\Lv_{\text{rm}}$</span>&#x7684;&#x7279;&#x5F81;&#x503C;/&#x5411;&#x91CF;&#xFF0C;<span class="mathjax-exps">$(0, \Dv^{1/2} \ev)$</span>&#x662F;<span class="mathjax-exps">$\Lv_{\text{sym}}$</span>&#x7684;&#x7279;&#x5F81;&#x503C;/&#x5411;&#x91CF;</p>
<p></p><div class="mathjax-exps">$$\begin{align*}         \Lv_{\text{rm}} \ev &amp; = \Dv^{-1} \Lv \ev = \Dv^{-1} 0 \ev = \zerov = 0 \ev \\         \Lv_{\text{sym}} \Dv^{1/2} \ev &amp; = \Dv^{-1/2} \Lv \Dv^{-1/2} \Dv^{1/2} \ev = \Dv^{-1/2} \Lv \ev = \Dv^{-1/2} 0 \ev = \zerov     \end{align*}$$</div><p></p>
<br>
<p><span class="mathjax-exps">$\Lv_{\text{sym}}$</span>&#x548C;<span class="mathjax-exps">$\Lv_{\text{rm}}$</span>&#x6700;&#x5C0F;&#x7279;&#x5F81;&#x503C;<span class="mathjax-exps">$0$</span>&#x7684;&#x51E0;&#x4F55;&#x91CD;&#x6570;&#x7B49;&#x4E8E;&#x56FE;&#x7684;&#x8FDE;&#x901A;&#x5206;&#x652F;&#x6570;</p>
<p>&#x8BC1;&#x660E;&#xFF1A;<span class="mathjax-exps">$\Lv_{\text{sym}}$</span>&#x548C;<span class="mathjax-exps">$\Lv_{\text{rm}}$</span>&#x4E0E;<span class="mathjax-exps">$\Lv$</span>&#x6709;&#x76F8;&#x540C;&#x7684;&#x5206;&#x5757;&#x5BF9;&#x89D2;&#x5F62;&#x5F0F;</p>
<ul>
<li><span class="mathjax-exps">$\Lv_{\text{rm}}$</span>&#x5BF9;&#x5E94;&#x4E8E;<span class="mathjax-exps">$0$</span>&#x7684;&#x7279;&#x5F81;&#x5411;&#x91CF;&#x7531;<span class="mathjax-exps">$\ev_{\Vcal_1}, \ldots, \ev_{\Vcal_K}$</span>&#x5F20;&#x6210;</li>
<li><span class="mathjax-exps">$\Lv_{\text{sym}}$</span>&#x5BF9;&#x5E94;&#x4E8E;<span class="mathjax-exps">$0$</span>&#x7684;&#x7279;&#x5F81;&#x5411;&#x91CF;&#x7531;<span class="mathjax-exps">$\Dv^{1/2} \ev_{\Vcal_1}, \ldots, \Dv^{1/2} \ev_{\Vcal_K}$</span>&#x5F20;&#x6210;</li>
</ul>
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</section></section><section data-notes lineno="220" class="slide " data-line="220" data-h="3" data-v="0">
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        <h5 class="title">&#x56FE;&#x5272;</h5>
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<p>&#x8BBE;<span class="mathjax-exps">$\Vcal = \Vcal_1 \times \cdots \times \Vcal_K$</span>&#xFF0C;&#x5B9A;&#x4E49;&#x56FE;&#x5272;&#x4E3A;&#x5212;&#x5206;&#x70B9;&#x96C6;&#x65F6;&#x9700;&#x7834;&#x574F;&#x7684;&#x8FB9;&#x6570;&#xFF1A;</p>
<div>
    $$
        \begin{align*}
            \cut (\Vcal_1, \ldots, \Vcal_K) = \frac{1}{2} \sum_{k \in [K]} | (u,v) \in \Ecal: u \in \Vcal_k, v \in \overline{\Vcal}_k |
        \end{align*}
    $$
</div>
<p>&#x4E00;&#x4E2A;&#x597D;&#x7684;&#x5212;&#x5206;(&#x5BF9;&#x70B9;&#x505A;&#x805A;&#x7C7B;)&#x5E94;&#x8BE5;&#x5C3D;&#x53EF;&#x80FD;&#x5C11;&#x7684;&#x7834;&#x574F;&#x8FB9;</p>
<p>&#x95EE;&#x9898;&#xFF1A;&#x53EF;&#x80FD;&#x53EA;&#x5212;&#x5206;&#x51FA;&#x4E00;&#x4E2A;&#x70B9;</p>
<br>
<p>&#x6BD4;&#x4F8B;&#x5272;&#x548C;&#x5F52;&#x4E00;&#x5316;&#x5272;</p>
<div>
    $$
        \begin{align*}
            \rcut (\{ \Vcal_k \}_{k \in [K]}) &amp; = \sum_{k \in [K]} \frac{\cut (\Vcal_k, \overline{\Vcal}_k)}{|\Vcal_k|} \\
            \ncut (\{ \Vcal_k \}_{k \in [K]}) &amp; = \sum_{k \in [K]} \frac{\cut (\Vcal_k, \overline{\Vcal}_k)}{\sum_{u \in \Vcal_k} d_u}
        \end{align*}
    $$
</div>
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</section><section><section data-notes lineno="265" class="slide " data-line="265" data-h="4" data-v="0">
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    <div class="title_hr"> 
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        <h5 class="title">&#x4E8C;&#x5206; &#x6BD4;&#x4F8B;&#x5272;</h5>
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<p>&#x8BBE;<span class="mathjax-exps">$K=2$</span>&#xFF0C;&#x5373;&#x5C06;&#x70B9;&#x96C6;&#x4E00;&#x5206;&#x4E3A;&#x4E8C;&#xFF0C;&#x5219;&#x4F18;&#x5316;&#x95EE;&#x9898;&#x4E3A;<span class="mathjax-exps">$\min_{\Acal \subseteq \Vcal} ~ \rcut (\Acal, \overline{\Acal})$</span></p>
<p>&#x5B9A;&#x4E49;&#x6307;&#x793A;&#x5411;&#x91CF;<span class="mathjax-exps">$\fv \in \Rbb^{|\Vcal|}$</span></p>
<div>
    $$
        \begin{align*}
            [\fv]_v = \begin{cases}
                \sqrt{|\overline{\Acal}| / |\Acal|} &amp; &#x82E5;~v \in \Acal \\
                - \sqrt{|\Acal| / |\overline{\Acal}|} &amp; &#x82E5;~v \not \in \Acal
            \end{cases}
        \end{align*}
    $$
</div>
<p>&#x6613;&#x77E5;</p>
<div>
    $$
        \begin{align*}
            &amp; \fv^\top \Lv \fv = \frac{1}{2} \sum_{u \in \Acal, v \not \in \Acal} ([\fv]_u - [\fv]_v)^2 + \frac{1}{2} \sum_{u \not \in \Acal, v \in \Acal} ([\fv]_u - [\fv]_v)^2 \\
            &amp; = \frac{1}{2} \sum_{u \in \Acal, v \not \in \Acal} \left( \sqrt{\frac{|\overline{\Acal}|}{|\Acal|}} + \sqrt{\frac{|\Acal|}{|\overline{\Acal}|}} \right)^2 + \frac{1}{2} \sum_{u \not \in \Acal, v \in \Acal} \left( \sqrt{\frac{|\Acal|}{|\overline{\Acal}|}} + \sqrt{\frac{|\overline{\Acal}|}{|\Acal|}} \right)^2
        \end{align*}
    $$
</div>
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</section><section vertical="true" data-notes lineno="309" class="slide " data-line="309" data-h="4" data-v="1">
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<div>
    $$
        \begin{align*}
            &amp; \fv^\top \Lv \fv = \frac{1}{2} \sum_{u \in \Acal, v \not \in \Acal} ([\fv]_u - [\fv]_v)^2 + \frac{1}{2} \sum_{u \not \in \Acal, v \in \Acal} ([\fv]_u - [\fv]_v)^2 \\
            &amp; = \frac{1}{2} \sum_{u \in \Acal, v \not \in \Acal} \left( \sqrt{\frac{|\overline{\Acal}|}{|\Acal|}} + \sqrt{\frac{|\Acal|}{|\overline{\Acal}|}} \right)^2 + \frac{1}{2} \sum_{u \not \in \Acal, v \in \Acal} \left( \sqrt{\frac{|\Acal|}{|\overline{\Acal}|}} + \sqrt{\frac{|\overline{\Acal}|}{|\Acal|}} \right)^2 \\
            &amp; = \frac{1}{2} \sum_{u \in \Acal, v \not \in \Acal} \left( \frac{|\overline{\Acal}|}{|\Acal|} + \frac{|\Acal|}{|\overline{\Acal}|} + 2 \right) + \frac{1}{2} \sum_{u \not \in \Acal, v \in \Acal} \left( \frac{|\overline{\Acal}|}{|\Acal|} + \frac{|\Acal|}{|\overline{\Acal}|} + 2 \right) \\
            &amp; = \cut (\Acal, \overline{\Acal}) \left( \frac{|\overline{\Acal}| + |\Acal|}{|\Acal|} + \frac{|\Acal| + |\overline{\Acal}|}{|\overline{\Acal}|} \right) \\
            &amp; = |\Vcal| \rcut (\Acal, \overline{\Acal})
        \end{align*}
    $$
</div>
<br>
<p>&#x8FD9;&#x8868;&#x660E;&#x76EE;&#x6807;&#x51FD;&#x6570;&#x53EF;&#x4EE5;&#x901A;&#x8FC7; Laplace &#x77E9;&#x9635;&#x8868;&#x793A;</p>
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        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
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<p><span class="mathjax-exps">$\fv$</span>&#x6EE1;&#x8DB3;</p>
<div>
    $$
        \begin{align*}
            \fv^\top \ev &amp; = \sum_{u \in \Acal} \sqrt{\frac{|\overline{\Acal}|}{|\Acal|}} - \sum_{u \not \in \Acal} \sqrt{\frac{|\Acal|}{|\overline{\Acal}|}} = |\Acal| \sqrt{\frac{|\overline{\Acal}|}{|\Acal|}} - |\overline{\Acal}| \sqrt{\frac{|\Acal|}{|\overline{\Acal}|}} = 0 \\
            \fv^\top \fv &amp; = \sum_{u \in \Vcal} [\fv]_u^2 = \sum_{u \in \Acal} [\fv]_u^2 + \sum_{u \not \in \Acal} [\fv]_u^2 = |\Acal| \frac{|\overline{\Acal}|}{|\Acal|} + |\overline{\Acal}| \frac{|\Acal|}{|\overline{\Acal}|} = |\Vcal|
        \end{align*}
    $$
</div>
<p>&#x7EFC;&#x4E0A;&#x4F18;&#x5316;&#x95EE;&#x9898;&#x4E3A;</p>
<div>
    $$
        \begin{align*}
            \min_{\Acal \subseteq \Vcal} ~ \fv^\top \Lv \fv \quad \st &amp; ~ [\fv]_v = \begin{cases}
                \sqrt{|\overline{\Acal}| / |\Acal|} &amp; &#x82E5;~v \in \Acal \\
                - \sqrt{|\Acal| / |\overline{\Acal}|} &amp; &#x82E5;~v \not \in \Acal
            \end{cases} \\
            &amp; ~ \fv^\top \ev = 0, ~ \fv^\top \fv = |\Vcal|
        \end{align*}
    $$
</div>
<p>&#x96BE;&#x70B9;&#xFF1A;&#x79BB;&#x6563;&#x4F18;&#x5316;&#xFF0C;NP-hard</p>
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        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
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</section><section vertical="true" data-notes lineno="388" class="slide " data-line="388" data-h="4" data-v="3">
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</div>
<p>&#x53BB;&#x6389;&#x79BB;&#x6563;&#x7EA6;&#x675F;&#xFF0C;&#x95EE;&#x9898;&#x653E;&#x677E;&#x4E3A;</p>
<div>
    $$
        \begin{align*}
            \min_{\fv} ~ \fv^\top \Lv \fv &amp; \quad \st ~ \fv^\top \ev = 0, ~ \fv^\top \fv = |\Vcal|
        \end{align*}
    $$
</div>
<p>&#x8BBE;<span class="mathjax-exps">$\Lv$</span>&#x7684;&#x7279;&#x5F81;&#x503C;&#xFF1A;<span class="mathjax-exps">$0 = \lambda_{|\Vcal|} \leq \cdots \leq \lambda_1$</span>&#xFF0C;&#x5BF9;&#x5E94;&#x7279;&#x5F81;&#x5411;&#x91CF;&#xFF1A;<span class="mathjax-exps">$\ev / \|\ev\| = \vv_{|\Vcal|}, \ldots, \vv_1$</span></p>
<p>&#x8BBE;<span class="mathjax-exps">$\fv = \alpha_1 \vv_1 + \cdots + \alpha_{|\Vcal|} \vv_{|\Vcal|}$</span>&#xFF0C;&#x7B2C;&#x4E00;&#x4E2A;&#x7EA6;&#x675F;<span class="mathjax-exps">$\fv^\top \ev = \alpha_{|\Vcal|} \|\ev\| = 0 \Rightarrow \alpha_{|\Vcal|} = 0$</span>&#xFF0C;&#x7B2C;&#x4E8C;&#x4E2A;&#x7EA6;&#x675F;<span class="mathjax-exps">$\fv^\top \fv = \alpha_1^2 + \cdots + \alpha_{|\Vcal|}^2 = |\Vcal|$</span></p>
<div>
    $$
        \begin{align*}
            \fv^\top \Lv \fv &amp; = (\alpha_1 \vv_1 + \cdots + \alpha_{|\Vcal|} \vv_{|\Vcal|})^\top (\alpha_1 \lambda_1 \vv_1 + \cdots + \alpha_{|\Vcal| } \lambda_{|\Vcal|} \vv_{|\Vcal|}) \\
            &amp; = \alpha_1^2 \lambda_1 + \cdots + \alpha_{|\Vcal|}^2 \lambda_{|\Vcal|} = \alpha_1^2 \lambda_1 + \cdots + \alpha_{|\Vcal|-1}^2 \lambda_{|\Vcal|-1} \\
            &amp; \geq \alpha_1^2 \lambda_{|\Vcal|-1} + \cdots + \alpha_{|\Vcal|-1}^2 \lambda_{|\Vcal|-1} = |\Vcal| \lambda_{|\Vcal|-1}
        \end{align*}
    $$
</div>
<p>&#x53D6;&#x7B49;&#x53F7;&#x6761;&#x4EF6;&#x4E3A;<span class="mathjax-exps">$\fv$</span>&#x6B63;&#x6BD4;&#x4E8E;&#x5012;&#x6570;&#x7B2C;&#x4E8C;&#x5C0F;&#x7279;&#x5F81;&#x503C;&#x5BF9;&#x5E94;&#x7684;&#x7279;&#x5F81;&#x5411;&#x91CF;</p>
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        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
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</section><section vertical="true" data-notes lineno="432" class="slide " data-line="432" data-h="4" data-v="4">
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<p>&#x4F18;&#x5316;&#x95EE;&#x9898;</p>
<div>
    $$
        \begin{align*}
            \min_{\fv \in\Rbb^{|\Vcal|}} ~ \fv^\top \Lv \fv &amp; \quad \st ~ \fv^\top \ev = 0, ~ \fv^\top \fv = |\Vcal|
        \end{align*}
    $$
</div>
<p>&#x7684;&#x6700;&#x4F18;&#x89E3;&#x4E3A;<span class="mathjax-exps">$\fv^\star = \sqrt{|\Vcal|} \vv_{|\Vcal|-1}$</span></p>
<br>
<p>&#x53CD;&#x63A8;&#x5212;&#x5206;</p>
<ul>
<li><span class="mathjax-exps">$v \in \Acal$</span>&#x5F53;&#x4E14;&#x4EC5;&#x5F53;<span class="mathjax-exps">$[\fv]_v \geq 0$</span></li>
<li>&#x5C06;<span class="mathjax-exps">$\{ [\fv]_1, \ldots, [\fv]_{|\Vcal|} \}$</span>&#x770B;&#x6210;<span class="mathjax-exps">$\Rbb$</span>&#x4E0A;&#x7684;<span class="mathjax-exps">$|\Vcal|$</span>&#x4E2A;&#x70B9;&#xFF0C;&#x805A;&#x6210;&#x4E24;&#x7C07;<span class="mathjax-exps">$\Ccal$</span>&#x3001;<span class="mathjax-exps">$\overline{\Ccal}$</span>&#xFF0C;<span class="mathjax-exps">$v \in \Acal$</span>&#x5F53;&#x4E14;&#x4EC5;&#x5F53;<span class="mathjax-exps">$[\fv]_v \in \Ccal$</span></li>
</ul>
<br>
<p>&#x540E;&#x8005;&#x5C31;&#x662F;<span class="mathjax-exps">$K = 2$</span>&#x7684;&#x672A;&#x5F52;&#x4E00;&#x5316;&#x8C31;&#x805A;&#x7C7B;</p>
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        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
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</section></section><section><section data-notes lineno="473" class="slide " data-line="473" data-h="5" data-v="0">
<div class="multi_column">
    <div class="title_hr"> 
        <hr class="hr_top">
        <h5 class="title">&#x591A;&#x5206; &#x6BD4;&#x4F8B;&#x5272;</h5>
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</div>
<p>&#x8BBE;<span class="mathjax-exps">$K &gt; 2$</span>&#xFF0C;&#x5219;&#x4F18;&#x5316;&#x95EE;&#x9898;&#x4E3A;<span class="mathjax-exps">$\min_{\Vcal_1, \ldots, \Vcal_K} ~ \rcut (\Vcal_1, \ldots, \Vcal_K)$</span></p>
<p>&#x5B9A;&#x4E49;<span class="mathjax-exps">$K$</span>&#x4E2A;&#x6307;&#x793A;&#x5411;&#x91CF;<span class="mathjax-exps">$\hv_k \in \Rbb^{|\Vcal|}$</span></p>
<div>
    $$
        \begin{align*}
            [\hv_k]_v = \begin{cases}
                1 / \sqrt{|\Vcal_k|} &amp; &#x82E5;~v \in \Vcal_k \\
                0 &amp; &#x82E5;~v \not \in \Vcal_k
            \end{cases}
        \end{align*}
    $$
</div>
<p>&#x6613;&#x77E5;</p>
<div>
    $$
        \begin{align*}
            \hv_k^\top \Lv \hv_k &amp; = \frac{1}{2} \sum_{(u,v) \in \Ecal} ([\hv_k]_u - [\hv_k]_v)^2 \\
            &amp; = \frac{1}{2} \sum_{u \in \Vcal_k, v \not \in \Vcal_k} ([\hv_k]_u - [\hv_k]_v)^2 + \frac{1}{2} \sum_{u \not \in \Vcal_k, v \in \Vcal_k} ([\hv_k]_u - [\hv_k]_v)^2 \\
            &amp; = \frac{1}{2} \sum_{u \in \Vcal_k, v \not \in \Vcal_k} \frac{1}{|\Vcal_k|} + \frac{1}{2} \sum_{u \not \in \Vcal_k, v \in \Vcal_k} \frac{1}{|\Vcal_k|} = \frac{\cut (\Vcal_k, \overline{\Vcal}_k)}{|\Vcal_k|} 
        \end{align*}
    $$
</div>
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</section><section vertical="true" data-notes lineno="518" class="slide " data-line="518" data-h="5" data-v="1">
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<p>&#x4E8E;&#x662F;</p>
<div>
    $$
        \begin{align*}
            \rcut (\Vcal_1, \ldots, \Vcal_K) &amp; = \sum_{k \in [K]} \frac{\cut(\Vcal_k, \overline{\Vcal_k})}{|\Vcal_k|} = \sum_{k \in [K]} \hv_k^\top \Lv \hv_k \\
            &amp; = \sum_{k \in [K]} [\Hv^\top \Lv \Hv]_{kk} = \tr (\Hv^\top \Lv \Hv)
        \end{align*}
    $$
</div>
<p>&#x5176;&#x4E2D;<span class="mathjax-exps">$\Hv = [\hv_1, \ldots, \hv_K]$</span>&#xFF0C;&#x6CE8;&#x610F;<span class="mathjax-exps">$\Hv$</span>&#x7684;&#x5217;&#x76F8;&#x4E92;&#x6B63;&#x4EA4;&#x4E14;<span class="mathjax-exps">$\Hv^\top \Hv = \Iv$</span></p>
<br>
<p>&#x7EFC;&#x4E0A;&#x4F18;&#x5316;&#x95EE;&#x9898;&#x4E3A;</p>
<div>
    $$
        \begin{align*}
            \min_{\Vcal_1, \ldots, \Vcal_K} ~ \tr (\Hv^\top \Lv \Hv) \quad \st ~ \Hv^\top \Hv = \Iv, ~ [\hv_k]_v = \begin{cases}
                1 / \sqrt{|\Vcal_k|} &amp; &#x82E5;~v \in \Vcal_k \\
                0 &amp; &#x82E5;~v \not \in \Vcal_k
            \end{cases} \\
        \end{align*}
    $$
</div>
<p>&#x96BE;&#x70B9;&#xFF1A;&#x79BB;&#x6563;&#x4F18;&#x5316;&#xFF0C;NP-hard</p>
<div class="footer">
    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x8C31;&#x56FE;&#x7406;&#x8BBA;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section><section vertical="true" data-notes lineno="566" class="slide " data-line="566" data-h="5" data-v="2">
<div class="multi_column">
    <div class="title_hr"> 
        <hr class="hr_top">
        <h5 class="title">&#x591A;&#x5206; &#x6BD4;&#x4F8B;&#x5272; &#x6C42;&#x89E3;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>&#x53BB;&#x6389;&#x79BB;&#x6563;&#x7EA6;&#x675F;&#xFF0C;&#x95EE;&#x9898;&#x653E;&#x677E;&#x4E3A;</p>
<div>
    $$
        \begin{align*}
            \min_{\Hv \in \Rbb^{|\Vcal| \times K}} ~ \tr (\Hv^\top \Lv \Hv) \quad \st ~ \Hv^\top \Hv = \Iv
        \end{align*}
    $$
</div>
<p>&#x8BBE;<span class="mathjax-exps">$\Lv = \sum_{i \in [|\Vcal|]} \lambda_i \vv_i \vv_i^\top = \Vv \Lambdav \Vv^\top$</span>&#xFF0C;&#x8BB0;<span class="mathjax-exps">$\Hv = \Vv \Qv$</span></p>
<ul>
<li><span class="mathjax-exps">$\Hv^\top \Lv \Hv = \Qv^\top \Vv^\top \Vv \Lambdav \Vv^\top \Vv \Qv = \Qv^\top \Lambdav \Qv$</span></li>
<li><span class="mathjax-exps">$\Hv^\top \Hv = \Qv^\top \Vv^\top \Vv \Qv = \Qv^\top \Qv$</span></li>
</ul>
<p>&#x5C06;<span class="mathjax-exps">$\Qv$</span>&#x6269;&#x5145;&#x4E3A;&#x6B63;&#x4EA4;&#x77E9;&#x9635;<span class="mathjax-exps">$\widetilde{\Qv} = [\Qv, \overline{\Qv}]$</span>&#xFF0C;&#x8003;&#x8651;&#x95EE;&#x9898;</p>
<div>
    $$
        \begin{align*}
            \min_{\widetilde{\Qv} \in \Rbb^{|\Vcal| \times |\Vcal|}} &amp; ~ \tr (\widetilde{\Qv}^\top \Lambdav \widetilde{\Qv}) = \tr (\Qv^\top \Lambdav \Qv) + \tr (\overline{\Qv}^\top \Lambdav \overline{\Qv}) \\
            \st &amp; ~ \widetilde{\Qv}^\top \widetilde{\Qv} = \begin{bmatrix} \Qv^\top \Qv \\ &amp; \overline{\Qv}^\top \overline{\Qv} \end{bmatrix} = \Iv
        \end{align*}
    $$
</div>
<div class="footer">
    <hr class="hr_bottom">
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        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x8C31;&#x56FE;&#x7406;&#x8BBA;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section><section vertical="true" data-notes lineno="610" class="slide " data-line="610" data-h="5" data-v="3">
<div class="multi_column">
    <div class="title_hr"> 
        <hr class="hr_top">
        <h5 class="title">&#x591A;&#x5206; &#x6BD4;&#x4F8B;&#x5272; &#x6C42;&#x89E3;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p><span class="mathjax-exps">$\widetilde{\Qv} = [\Qv, \overline{\Qv}]$</span>&#xFF0C;&#x8003;&#x8651;&#x95EE;&#x9898;</p>
<div>
    $$
        \begin{align*}
            \min_{\widetilde{\Qv} \in \Rbb^{|\Vcal| \times |\Vcal|}} ~ \tr (\widetilde{\Qv}^\top \Lambdav \widetilde{\Qv}) \quad \st ~ \widetilde{\Qv}^\top \widetilde{\Qv} = \Iv
        \end{align*}
    $$
</div>
<ul>
<li><span class="mathjax-exps">$\tr (\widetilde{\Qv}^\top \Lambdav \widetilde{\Qv}) = \sum_i \qv_i^\top \Lambdav \qv_i = \sum_{i,j} \lambda_j \widetilde{q}_{ij}^2$</span></li>
<li><span class="mathjax-exps">$\Sv = \widetilde{\Qv} \circ \widetilde{\Qv} = (\widetilde{q}_{ij}^2)$</span>&#x662F;&#x4E00;&#x4E2A;&#x53CC;&#x968F;&#x673A;&#x77E9;&#x9635;&#xFF0C;&#x884C;&#x548C;&#x5217;&#x548C;&#x5747;&#x4E3A;<span class="mathjax-exps">$1$</span></li>
</ul>
<div>
    $$
        \begin{align*}
            \min_{\widetilde{\Qv}^\top \widetilde{\Qv} = \Iv} \tr (\widetilde{\Qv}^\top \Lambdav \widetilde{\Qv}) = \min_{\widetilde{\Qv}^\top \widetilde{\Qv} = \Iv} \sum_{i,j} \lambda_j \widetilde{q}_{ij}^2 \geq \min_{\Sv \in \Scal} \sum_{i,j} \lambda_j s_{ij}
        \end{align*}
    $$
</div>
<p>&#x5176;&#x4E2D;<span class="mathjax-exps">$\Scal$</span>&#x662F;&#x53CC;&#x968F;&#x673A;&#x77E9;&#x9635;&#x96C6;&#x5408;&#xFF0C;&#x4E0A;&#x5F0F;&#x6700;&#x53F3;&#x7AEF;&#x662F;&#x7EBF;&#x6027;&#x89C4;&#x5212;&#xFF0C;&#x6240;&#x4EE5;&#x6700;&#x4F18;&#x89E3;&#x4E3A;&#x53EF;&#x884C;&#x57DF;<span class="mathjax-exps">$\Scal$</span>&#x7684;&#x6781;&#x70B9;&#xFF0C;&#x7531; Birkhoff&#x2013;von Neumann &#x5B9A;&#x7406;&#x77E5;<span class="mathjax-exps">$\Scal$</span>&#x7684;&#x6781;&#x70B9;&#x662F;&#x7F6E;&#x6362;&#x77E9;&#x9635;&#xFF0C;&#x800C;&#x7F6E;&#x6362;&#x77E9;&#x9635;&#x662F;&#x6B63;&#x4EA4;&#x77E9;&#x9635;&#xFF0C;&#x56E0;&#x6B64;&#x4E0A;&#x5F0F;&#x4E0D;&#x7B49;&#x53F7;&#x53EF;&#x4EE5;&#x53D6;&#x7B49;&#x53F7;&#xFF0C;&#x539F;&#x95EE;&#x9898;&#x53EA;&#x53D6;<span class="mathjax-exps">$\widetilde{\Qv}$</span>&#x7684;&#x524D;<span class="mathjax-exps">$K$</span>&#x5217;&#xFF0C;&#x6545;&#x539F;&#x95EE;&#x9898;&#x6700;&#x4F18;&#x89E3;&#x662F;&#x6700;&#x5C0F;&#x7684;<span class="mathjax-exps">$K$</span>&#x4E2A;&#x7279;&#x5F81;&#x503C;&#x5BF9;&#x5E94;&#x7684;&#x7279;&#x5F81;&#x5411;&#x91CF;</p>
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        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x8C31;&#x56FE;&#x7406;&#x8BBA;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section><section vertical="true" data-notes lineno="651" class="slide " data-line="651" data-h="5" data-v="4">
<div class="multi_column">
    <div class="title_hr"> 
        <hr class="hr_top">
        <h5 class="title">&#x591A;&#x5206; &#x6BD4;&#x4F8B;&#x5272; &#x6C42;&#x89E3;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>&#x95EE;&#x9898;</p>
<div>
    $$
        \begin{align*}
            \min_{\Hv \in \Rbb^{|\Vcal| \times K}} ~ \tr (\Hv^\top \Lv \Hv) \quad \st ~ \Hv^\top \Hv = \Iv
        \end{align*}
    $$
</div>
<p>&#x7684;&#x6700;&#x4F18;&#x89E3;&#x4E3A;<span class="mathjax-exps">$\Lv$</span>&#x6700;&#x5C0F;&#x7684;<span class="mathjax-exps">$K$</span>&#x4E2A;&#x7279;&#x5F81;&#x503C;&#x5BF9;&#x5E94;&#x7684;&#x7279;&#x5F81;&#x5411;&#x91CF;&#xFF0C;&#x4E0D;&#x59A8;&#x8BB0;&#x4E3A;<span class="mathjax-exps">$\vv_1, \ldots, \vv_K$</span></p>
<br>
<p>&#x53CD;&#x63A8;&#x5212;&#x5206;&#xFF1A;&#x5C06;<span class="mathjax-exps">$\Vv= [\vv_1, \ldots, \vv_K] \in \Rbb^{|\Vcal| \times K}$</span>&#x770B;&#x4F5C;<span class="mathjax-exps">$|\Vcal|$</span>&#x4E2A;<span class="mathjax-exps">$K$</span>&#x7EF4;&#x70B9;&#xFF0C;&#x5C06;&#x5176;&#x805A;&#x6210;<span class="mathjax-exps">$K$</span>&#x7C07;<span class="mathjax-exps">$\Ccal_1, \ldots, \Ccal_K$</span>&#xFF0C;<span class="mathjax-exps">$v \in \Vcal_k$</span>&#x5F53;&#x4E14;&#x4EC5;&#x5F53;<span class="mathjax-exps">$[\Vv]_v \in \Ccal_k$</span></p>
<br>
<p>&#x8FD9;&#x5C31;&#x662F;&#x4E00;&#x822C;&#x5F62;&#x5F0F;&#x7684;&#x672A;&#x5F52;&#x4E00;&#x5316;&#x8C31;&#x805A;&#x7C7B;</p>
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        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x8C31;&#x56FE;&#x7406;&#x8BBA;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section></section><section><section data-notes lineno="689" class="slide " data-line="689" data-h="6" data-v="0">
<div class="multi_column">
    <div class="title_hr"> 
        <hr class="hr_top">
        <h5 class="title">&#x4E8C;&#x5206; &#x5F52;&#x4E00;&#x5316;&#x5272;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>&#x4F18;&#x5316;&#x95EE;&#x9898;&#x4E3A;<span class="mathjax-exps">$\min_{\Acal \subseteq \Vcal} ~ \ncut (\Acal, \overline{\Acal})$</span></p>
<p>&#x5B9A;&#x4E49;&#x6307;&#x793A;&#x5411;&#x91CF;<span class="mathjax-exps">$\fv \in \Rbb^{|\Vcal|}$</span></p>
<div>
    $$
        \begin{align*}
            [\fv]_v = \begin{cases}
                \sqrt{ \vol(\overline{\Acal}) / \vol(\Acal) } &amp; &#x82E5;~v \in \Acal \\
                - \sqrt{ \vol(\Acal) / \vol(\overline{\Acal}) } &amp; &#x82E5;~v \not \in \Acal
            \end{cases}
        \end{align*}
    $$
</div>
<p>&#x6613;&#x77E5;</p>
<div>
    $$
        \begin{align*}
            \fv^\top \Lv \fv &amp; = \frac{1}{2} \sum_{u \in \Acal, v \not \in \Acal} \left( \sqrt{\frac{\vol(\overline{\Acal})}{\vol(\Acal)}} + \sqrt{\frac{\vol(\Acal)}{\vol(\overline{\Acal})}} \right)^2  \\
            &amp; \qquad + \frac{1}{2} \sum_{u \not \in \Acal, v \in \Acal} \left( \sqrt{\frac{\vol(\Acal)}{\vol(\overline{\Acal})}} + \sqrt{\frac{\vol(\overline{\Acal})}{\vol(\Acal)}} \right)^2
        \end{align*}
    $$
</div>
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        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x8C31;&#x56FE;&#x7406;&#x8BBA;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section><section vertical="true" data-notes lineno="733" class="slide " data-line="733" data-h="6" data-v="1">
<div class="multi_column">
    <div class="title_hr"> 
        <hr class="hr_top">
        <h5 class="title">&#x4E8C;&#x5206; &#x5F52;&#x4E00;&#x5316;&#x5272;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<div>
    $$
        \begin{align*}
            \fv^\top \Lv \fv &amp; = \frac{1}{2} \sum_{u \in \Acal, v \not \in \Acal} \left( \frac{\vol(\overline{\Acal})}{\vol(\Acal)} + \frac{\vol(\Acal)}{\vol(\overline{\Acal})} + 2 \right)  \\
            &amp; \qquad + \frac{1}{2} \sum_{u \not \in \Acal, v \in \Acal} \left( \frac{\vol(\Acal)}{\vol(\overline{\Acal})} + \frac{\vol(\overline{\Acal})}{\vol(\Acal)} + 2 \right) \\
            &amp; = \cut (\Acal, \overline{\Acal}) \left( \frac{\vol(\Acal) + \vol(\overline{\Acal})}{\vol(\overline{\Acal})} + \frac{\vol(\overline{\Acal}) + \vol(\Acal)}{\vol(\Acal)}\right) \\
            &amp; = \vol(\Vcal) \cut (\Acal, \overline{\Acal}) \left( \frac{1}{\vol(\overline{\Acal})} + \frac{1}{\vol(\Acal)}\right) \\
            &amp; = \vol(\Vcal) \ncut (\Acal, \overline{\Acal})
        \end{align*}
    $$
</div>
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        <h6 class="bottom_center">&#x8C31;&#x56FE;&#x7406;&#x8BBA;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
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</div>
</section><section vertical="true" data-notes lineno="763" class="slide " data-line="763" data-h="6" data-v="2">
<div class="multi_column">
    <div class="title_hr"> 
        <hr class="hr_top">
        <h5 class="title">&#x4E8C;&#x5206; &#x5F52;&#x4E00;&#x5316;&#x5272;</h5>
    </div>
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</div>
<div>
    $$
        \begin{align*}
            (D \fv)^\top \ev &amp; = \sum_{u \in \Acal} d_u \sqrt{\frac{\vol(\overline{\Acal})}{\vol(\Acal)}} - \sum_{u \not \in \Acal} d_u \sqrt{\frac{\vol(\Acal)}{\vol(\overline{\Acal})}} \\
            &amp; = \vol(\Acal) \sqrt{\frac{\vol(\overline{\Acal})}{\vol(\Acal)}} - \vol(\overline{\Acal}) \sqrt{\frac{\vol(\Acal)}{\vol(\overline{\Acal})}} \\
            &amp; = 0 \\
            \fv^\top \Dv \fv &amp; = \sum_{u \in \Acal} d_u [\fv]_u^2 + \sum_{u \not \in \Acal} d_u [\fv]_u^2 = \sum_{u \in \Acal} d_u \frac{\vol(\overline{\Acal})}{\vol(\Acal)} + \sum_{u \not \in \Acal} d_u \frac{\vol(\Acal)}{\vol(\overline{\Acal})} \\
            &amp; = \vol(\Acal) \frac{\vol(\overline{\Acal})}{\vol(\Acal)} + \vol(\overline{\Acal}) \frac{\vol(\Acal)}{\vol(\overline{\Acal})} \\
            &amp; = \vol(\Vcal)
        \end{align*}
    $$
</div>
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        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x8C31;&#x56FE;&#x7406;&#x8BBA;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
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</div>
</section><section vertical="true" data-notes lineno="794" class="slide " data-line="794" data-h="6" data-v="3">
<div class="multi_column">
    <div class="title_hr"> 
        <hr class="hr_top">
        <h5 class="title">&#x4E8C;&#x5206; &#x5F52;&#x4E00;&#x5316;&#x5272;</h5>
    </div>
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</div>
<p>&#x7EFC;&#x4E0A;&#x4F18;&#x5316;&#x95EE;&#x9898;&#x4E3A;</p>
<div>
    $$
        \begin{align*}
            \min_{\Acal \subseteq \Vcal} ~ \fv^\top \Lv \fv \quad \st &amp; ~ [\fv]_v = \begin{cases}
                \sqrt{ \vol(\overline{\Acal}) / \vol(\Acal) } &amp; &#x82E5;~v \in \Acal \\
                - \sqrt{ \vol(\Acal) / \vol(\overline{\Acal}) } &amp; &#x82E5;~v \not \in \Acal
            \end{cases}~~~~~~ \\
            &amp; ~ (D \fv)^\top \ev = 0, ~ \fv^\top \Dv \fv = \vol(\Vcal)
        \end{align*}
    $$
</div>
<br>
<p>&#x96BE;&#x70B9;&#xFF1A;&#x79BB;&#x6563;&#x4F18;&#x5316;&#xFF0C;NP-hard</p>
<br>
<p>&#x5C06;&#x95EE;&#x9898;&#x653E;&#x677E;&#x4E3A;</p>
<div>
    $$
        \begin{align*}
            \min_{\fv \in \Rbb^{|\Vcal|}} ~ \fv^\top \Lv \fv \quad \st ~ (D \fv)^\top \ev = 0, ~ \fv^\top \Dv \fv = \vol(\Vcal)
        \end{align*}
    $$
</div>
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        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x8C31;&#x56FE;&#x7406;&#x8BBA;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
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</section><section vertical="true" data-notes lineno="842" class="slide " data-line="842" data-h="6" data-v="4">
<div class="multi_column">
    <div class="title_hr"> 
        <hr class="hr_top">
        <h5 class="title">&#x4E8C;&#x5206; &#x5F52;&#x4E00;&#x5316;&#x5272; &#x6C42;&#x89E3;</h5>
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    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>&#x5C06;&#x95EE;&#x9898;&#x653E;&#x677E;&#x4E3A;</p>
<div>
    $$
        \begin{align*}
            \min_{\fv \in \Rbb^{|\Vcal|}} ~ \fv^\top \Lv \fv \quad \st ~ (D \fv)^\top \ev = 0, ~ \fv^\top \Dv \fv = \vol(\Vcal)
        \end{align*}
    $$
</div>
<br>
<p>&#x8BB0;<span class="mathjax-exps">$\gv = \Dv^{1/2} \fv$</span>&#xFF0C;&#x95EE;&#x9898;&#x53EF;&#x8FDB;&#x4E00;&#x6B65;&#x5199;&#x4E3A;</p>
<div>
    $$
        \begin{align*}
            \min_{\fv \in \Rbb^{|\Vcal|}} ~ \gv^\top \Dv^{-1/2} \Lv \Dv^{-1/2} \gv \quad \st ~ \gv^\top \Dv^{1/2} \ev = 0, ~ \gv^\top \gv = \vol(\Vcal)
        \end{align*}
    $$
</div>
<br>
<p>&#x6CE8;&#x610F;<span class="mathjax-exps">$\Dv^{-1/2} \Lv \Dv^{-1/2} = \Lv_{\text{sym}}$</span>&#xFF0C;<span class="mathjax-exps">$\Dv^{1/2} \ev$</span>&#x662F;&#x5176;&#x6700;&#x5C0F;&#x7684;&#x7279;&#x5F81;&#x5411;&#x91CF;&#xFF0C;&#x56E0;&#x6B64;&#x6700;&#x4F18;&#x89E3;<span class="mathjax-exps">$\gv^\star$</span>&#x662F;&#x5176;&#x5012;&#x6570;&#x7B2C;&#x4E8C;&#x5C0F;&#x7684;&#x7279;&#x5F81;&#x5411;&#x91CF;</p>
<br>
<p>&#x8FDB;&#x4E00;&#x6B65;<span class="mathjax-exps">$\fv^\star = \Dv^{-1/2} \gv^\star$</span>&#x662F;<span class="mathjax-exps">$\Lv_{\text{rm}}$</span>&#x5012;&#x6570;&#x7B2C;&#x4E8C;&#x5C0F;&#x7684;&#x7279;&#x5F81;&#x5411;&#x91CF;</p>
<br>
<p>&#x5BF9;<span class="mathjax-exps">$\fv^\star$</span>&#x505A;&#x805A;&#x7C7B;&#x53CD;&#x63A8;&#x5212;&#x5206;&#x5C31;&#x662F;<span class="mathjax-exps">$K=2$</span>&#x7684;&#x5F52;&#x4E00;&#x5316;&#x8C31;&#x805A;&#x7C7B;</p>
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        <h6 class="bottom_center">&#x8C31;&#x56FE;&#x7406;&#x8BBA;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section></section><section><section data-notes lineno="894" class="slide " data-line="894" data-h="7" data-v="0">
<div class="multi_column">
    <div class="title_hr"> 
        <hr class="hr_top">
        <h5 class="title">&#x591A;&#x5206; &#x5F52;&#x4E00;&#x5316;&#x5272;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>&#x8BBE;<span class="mathjax-exps">$K &gt; 2$</span>&#xFF0C;&#x5219;&#x4F18;&#x5316;&#x95EE;&#x9898;&#x4E3A;<span class="mathjax-exps">$\min_{\Vcal_1, \ldots, \Vcal_K} ~ \ncut (\Vcal_1, \ldots, \Vcal_K)$</span></p>
<p>&#x5B9A;&#x4E49;<span class="mathjax-exps">$K$</span>&#x4E2A;&#x6307;&#x793A;&#x5411;&#x91CF;<span class="mathjax-exps">$\hv_k \in \Rbb^{|\Vcal|}$</span></p>
<div>
    $$
        \begin{align*}
            [\hv_k]_v = \begin{cases}
                1 / \sqrt{\vol(\Vcal_k)} &amp; &#x82E5;~v \in \Vcal_k \\
                0 &amp; &#x82E5;~v \not \in \Vcal_k
            \end{cases}
        \end{align*}
    $$
</div>
<p>&#x6613;&#x77E5;</p>
<div>
    $$
        \begin{align*}
            \hv_k^\top \Lv \hv_k &amp; = \frac{1}{2} \sum_{(u,v) \in \Ecal} ([\hv_k]_u - [\hv_k]_v)^2 \\
            &amp; = \frac{1}{2} \sum_{u \in \Vcal_k, v \not \in \Vcal_k} ([\hv_k]_u - [\hv_k]_v)^2 + \frac{1}{2} \sum_{u \not \in \Vcal_k, v \in \Vcal_k} ([\hv_k]_u - [\hv_k]_v)^2 \\
            &amp; = \frac{1}{2} \sum_{u \in \Vcal_k, v \not \in \Vcal_k} \frac{1}{\vol(\Vcal_k)} + \frac{1}{2} \sum_{u \not \in \Vcal_k, v \in \Vcal_k} \frac{1}{\vol(\Vcal_k)} = \frac{\cut (\Vcal_k, \overline{\Vcal}_k)}{\vol(\Vcal_k)}
        \end{align*}
    $$
</div>
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        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x8C31;&#x56FE;&#x7406;&#x8BBA;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
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</section><section vertical="true" data-notes lineno="939" class="slide " data-line="939" data-h="7" data-v="1">
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    <div class="title_hr"> 
        <hr class="hr_top">
        <h5 class="title">&#x591A;&#x5206; &#x5F52;&#x4E00;&#x5316;&#x5272;</h5>
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<p>&#x4E8E;&#x662F;</p>
<div>
    $$
        \begin{align*}
            \ncut (\Vcal_1, \ldots, \Vcal_K) &amp; = \sum_{k \in [K]} \frac{\cut(\Vcal_k, \overline{\Vcal_k})}{\vol(\Vcal_k)} = \sum_{k \in [K]} \hv_k^\top \Lv \hv_k \\
            &amp; = \sum_{k \in [K]} [\Hv^\top \Lv \Hv]_{kk} = \tr (\Hv^\top \Lv \Hv)
        \end{align*}
    $$
</div>
<p>&#x5176;&#x4E2D;<span class="mathjax-exps">$\Hv = [\hv_1, \ldots, \hv_K]$</span>&#xFF0C;&#x4E14;&#x6709;<span class="mathjax-exps">$\hv_k \Dv \hv_k = \sum_{v \in \Vcal} d_v / \vol(\Vcal_k) = 1$</span>&#xFF0C;&#x5199;&#x6210;&#x77E9;&#x9635;&#x7684;&#x5F62;&#x5F0F;<span class="mathjax-exps">$\Hv^\top \Dv \Hv = \Iv$</span></p>
<br>
<p>&#x7EFC;&#x4E0A;&#x4F18;&#x5316;&#x95EE;&#x9898;&#x4E3A;</p>
<div>
    $$
        \begin{align*}
            \min_{\Vcal_1, \ldots, \Vcal_K} ~ \tr (\Hv^\top \Lv \Hv) \quad \st &amp; ~ [\hv_k]_v = \begin{cases}
                1 / \sqrt{\vol(\Vcal_k)} &amp; &#x82E5;~v \in \Vcal_k \\
                0 &amp; &#x82E5;~v \not \in \Vcal_k
            \end{cases} \\
            &amp; ~ \Hv^\top \Dv \Hv = \Iv
        \end{align*}
    $$
</div>
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        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x8C31;&#x56FE;&#x7406;&#x8BBA;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section><section vertical="true" data-notes lineno="986" class="slide " data-line="986" data-h="7" data-v="2">
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    <div class="title_hr"> 
        <hr class="hr_top">
        <h5 class="title">&#x591A;&#x5206; &#x5F52;&#x4E00;&#x5316;&#x5272; &#x6C42;&#x89E3;</h5>
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<p>&#x53BB;&#x6389;&#x79BB;&#x6563;&#x7EA6;&#x675F;&#xFF0C;&#x95EE;&#x9898;&#x653E;&#x677E;&#x4E3A;</p>
<div>
    $$
        \begin{align*}
            \min_{\Hv \in \Rbb^{|\Vcal| \times K}} ~ \tr (\Hv^\top \Lv \Hv) \quad \st ~ \Hv^\top \Dv \Hv = \Iv
        \end{align*}
    $$
</div>
<br>
<p>&#x8BB0;<span class="mathjax-exps">$\Tv = \Dv^{1/2} \Hv$</span>&#xFF0C;&#x95EE;&#x9898;&#x53EF;&#x8FDB;&#x4E00;&#x6B65;&#x5199;&#x4E3A;</p>
<div>
    $$
        \begin{align*}
            \min_{\Tv \in \Rbb^{|\Vcal| \times K}} ~ \Tv^\top \underbrace{\Dv^{-1/2} \Lv \Dv^{-1/2}}_{\Lv_{\text{sym}}} \Tv \quad \st ~ \Tv^\top \Tv = \Iv
        \end{align*}
    $$
</div>
<p>&#x6700;&#x4F18;&#x89E3;<span class="mathjax-exps">$\Tv^\star$</span>&#x662F;<span class="mathjax-exps">$\Lv_{\text{sym}}$</span>&#x6700;&#x5C0F;<span class="mathjax-exps">$K$</span>&#x4E2A;&#x7279;&#x5F81;&#x503C;&#x5BF9;&#x5E94;&#x7684;&#x7279;&#x5F81;&#x5411;&#x91CF;&#xFF0C;&#x8FDB;&#x4E00;&#x6B65;&#x539F;&#x95EE;&#x9898;&#x6700;&#x4F18;&#x89E3;<span class="mathjax-exps">$\Hv^\star = \Dv^{-1/2} \Tv^\star$</span>&#x662F;<span class="mathjax-exps">$\Lv_{\text{rm}}$</span>&#x6700;&#x5C0F;<span class="mathjax-exps">$K$</span>&#x4E2A;&#x7279;&#x5F81;&#x503C;&#x5BF9;&#x5E94;&#x7684;&#x7279;&#x5F81;&#x5411;&#x91CF;</p>
<br>
<p>&#x540C;&#x6837;&#x5C06;<span class="mathjax-exps">$K$</span>&#x4E2A;&#x7279;&#x5F81;&#x5411;&#x91CF;&#x805A;&#x6210;<span class="mathjax-exps">$K$</span>&#x7C07;&#x53EF;&#x4EE5;&#x53CD;&#x63A8;&#x51FA;&#x5212;&#x5206;&#xFF0C;&#x5C31;&#x662F;&#x4E00;&#x822C;&#x5F62;&#x5F0F;&#x7684;&#x5F52;&#x4E00;&#x5316;&#x8C31;&#x805A;&#x7C7B;</p>
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        <h6 class="bottom_center">&#x8C31;&#x56FE;&#x7406;&#x8BBA;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
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